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(section.3) cvn /Title (The polynomial sequences a\(n;r\), product sequences b\(n;r\) and Riordan arrays) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.4) cvn /Title (a\(n;r\) and Chebyshev polynomials) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.5) cvn /Title (b\(n;r\) and Chebyshev polynomials) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.6) cvn /Title (From b\(n;r\) to a\(n;r\)) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.7) cvn /Title (A Chebyshev product result) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.8) cvn /Title (Additional Riordan array factorizations) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.9) cvn /Title (A left inverse) /OUT pdfmark end 800 -4000 a 800 -4000 a SDict begin [ /Count -0 /Dest (section.10) cvn /Title (The matrix \(1-x1-x4,x1+x2\) and its 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Fq(;)221 b(x)p Fs(\))296 b(+)f Fq(x)p Fn(P)109 b Fs(\()p Fq(n)296 b Fn(\000)f Fs(2)p Fq(;)221 b(x)p Fs(\))p Fq(;)1523 b(n)370 b(>)e Fs(1)p Black 26150 70521 a(10)p Black eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(then)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)19908 -635 y Fq(a)p Fs(\()p Fq(n;)221 b(r)36 b Fs(\))370 b(=)e Fq(r)25955 -1183 y Fl(b)26506 -1545 y Fj(n)p 26507 -1384 512 40 v 26560 -836 a Fk(2)27151 -1183 y Fl(c)27624 -635 y Fn(P)28879 -2508 y 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800 60511 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark end 800 60511 a Black 2423 x Fg(Theorem)499 b(5.)p Black 554 w Fh(L)-66 b(et)464 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))466 b Fh(b)-66 b(e)464 b(the)g(se)-66 b(quenc)g(e)464 b(of)g(p)-66 b(olynomials)465 b(with)f(gener)-66 b(ating)463 b(function)18399 66621 y Fq(g)19022 66820 y Fp(a)19577 66621 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)26621 65722 y(1)296 b(+)e Fq(x)i Fn(\000)f Fq(x)31975 65240 y Fo(2)p 24415 66315 10292 54 v 24415 67532 a Fs(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)31314 67148 y Fo(2)32135 67532 y Fs(+)g Fq(x)34181 67148 y Fo(4)34840 66621 y Fq(:)p Black 26150 70521 a Fs(11)p Black eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 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Fs(\(6\))800 27376 y(By)434 b(direct)f(insp)36 b(ection,)434 b(the)f(\014rst)g(terms)g(of)h(the)f (sequence)h Fq(A)31701 27575 y Fp(n)32696 27376 y Fs(=)369 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))434 b(are)14380 30036 y(1)p Fq(;)221 b Fs(1)p Fq(;)g Fs(1)298 b(+)d Fq(r)-36 b(;)221 b Fs(2)296 b(+)f Fq(r)-36 b(;)221 b Fs(1)296 b(+)f(3)p Fq(r)332 b Fs(+)294 b Fq(r)29365 29487 y Fo(2)29891 30036 y Fq(;)221 b Fs(3)297 b(+)d(4)p Fq(r)332 b Fs(+)295 b Fq(r)36223 29487 y Fo(2)36749 30036 y Fq(;)221 b(:)g(:)g(:)r(:)800 32696 y Fs(W)-108 b(e)626 b(no)-36 b(w)626 b(set)f Fq(B)9344 32895 y Fp(n)10667 32696 y Fs(=)696 b Fq(A)13350 32895 y Fp(n)13976 32696 y Fq(A)14951 32895 y Fp(n)p Fo(+1)17405 32696 y Fs(and)626 b Fq(T)20889 32895 y Fp(n)22211 32696 y Fs(=)696 b Fq(A)24894 32895 y Fp(n)25520 32696 y Fq(A)26495 32895 y Fp(n)p Fo(+3)28324 32696 y Fs(,)674 b(and)625 b(try)h(to)g(determine)f(the)g (corresp)36 b(onding)800 34301 y(generating)434 b(functions.)578 b(F)-108 b(rom)433 b(recurrence)g(\()p 0 .5 0 TeXcolorrgb 23479 34301 a SDict begin H.S end 23479 34301 a 0 .5 0 TeXcolorrgb Fs(6)p 0 .5 0 TeXcolorrgb 24129 33445 a SDict begin H.R end 24129 33445 a 24129 34301 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 24129 34301 a Black Fs(\),)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)18046 36961 y Fq(A)19021 37160 y Fp(n)p Fo(+4)21957 36961 y Fs(=)1106 b(\()p Fq(r)332 b Fs(+)295 b(2\))p Fq(A)28937 37160 y Fp(n)p Fo(+2)31061 36961 y Fn(\000)g Fq(A)33364 37160 y Fp(n)18046 38898 y Fq(A)19021 39097 y Fp(n)p Fo(+5)21957 38898 y Fs(=)1106 b(\()p Fq(r)332 b Fs(+)295 b(2\))p Fq(A)28937 39097 y Fp(n)p Fo(+3)31061 38898 y Fn(\000)g Fq(A)33364 39097 y Fp(n)p Fo(+1)35192 38898 y Fq(:)800 41558 y Fs(Multiplying,)434 b(w)-36 b(e)434 b(obtain)4412 44218 y Fq(A)5387 44417 y Fp(n)p Fo(+4)7216 44218 y Fq(A)8191 44417 y Fp(n)p Fo(+5)10388 44218 y Fs(=)369 b(\()p Fq(r)331 b Fs(+)295 b(2\))15655 43669 y Fo(2)16181 44218 y Fq(A)17156 44417 y Fp(n)p Fo(+2)18985 44218 y Fq(A)19960 44417 y Fp(n)p Fo(+3)22083 44218 y Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(A)28273 44417 y Fp(n)p Fo(+1)30102 44218 y Fq(A)31077 44417 y Fp(n)p Fo(+2)33200 44218 y Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(A)39390 44417 y Fp(n)40016 44218 y Fq(A)40991 44417 y Fp(n)p Fo(+3)43115 44218 y Fs(+)g Fq(A)45397 44417 y Fp(n)46023 44218 y Fq(A)46998 44417 y Fp(n)p Fo(+1)48827 44218 y Fq(:)800 46878 y Fs(This)434 b(can)f(b)36 b(e)434 b(written)f(as:)14580 45273 y SDict begin H.S end 14580 45273 a 14580 45273 a SDict begin 14.5 H.A end 14580 45273 a 14580 45273 a SDict begin [ /View [/XYZ H.V] /Dest (equation.7) cvn H.B /DEST pdfmark end 14580 45273 a 11501 49538 a Fq(B)12489 49737 y Fp(n)p Fo(+4)14687 49538 y Fs(=)369 b(\()p Fq(r)331 b Fs(+)295 b(2\))19954 48989 y Fo(2)20480 49538 y Fq(B)21468 49737 y Fp(n)p Fo(+2)23591 49538 y Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(B)29794 49737 y Fp(n)p Fo(+1)31918 49538 y Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(2\))p Fq(T)37894 49737 y Fp(n)38816 49538 y Fs(+)h Fq(B)41111 49737 y Fp(n)41737 49538 y Fq(:)9040 b Fs(\(7\))800 52197 y(No)-36 b(w)434 b(m)-36 b(ultiplying)434 b(equation)g(\()p 0 .5 0 TeXcolorrgb 16516 52197 a SDict begin H.S end 16516 52197 a 0 .5 0 TeXcolorrgb Fs(6)p 0 .5 0 TeXcolorrgb 17166 51341 a SDict begin H.R end 17166 51341 a 17166 52197 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 17166 52197 a Black Fs(\))g(b)-36 b(y)433 b Fq(A)20887 52396 y Fp(n)p Fo(+3)22716 52197 y Fs(,)h(w)-36 b(e)433 b(obtain)15180 54857 y Fq(A)16155 55056 y Fp(n)p Fo(+3)17984 54857 y Fq(A)18959 55056 y Fp(n)p Fo(+4)21156 54857 y Fs(=)369 b(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(A)27398 55056 y Fp(n)p Fo(+2)29227 54857 y Fq(A)30202 55056 y Fp(n)p Fo(+3)32325 54857 y Fn(\000)h Fq(A)34629 55056 y Fp(n)35255 54857 y Fq(A)36230 55056 y Fp(n)p Fo(+3)38059 54857 y Fq(;)800 57517 y Fs(or)2390 55912 y SDict begin H.S end 2390 55912 a 2390 55912 a SDict begin 14.5 H.A end 2390 55912 a 2390 55912 a SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark end 2390 55912 a 19479 59122 a Fq(B)20467 59321 y Fp(n)p Fo(+3)22664 59122 y Fs(=)369 b(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(B)28919 59321 y Fp(n)p Fo(+2)31043 59122 y Fn(\000)g Fq(T)33133 59321 y Fp(n)33760 59122 y Fq(:)17017 b Fs(\(8\))800 61330 y(W)-108 b(e)346 b(can)f(no)-36 b(w)346 b(use)g(the)f(metho)36 b(d)345 b(of)h(co)36 b(e\016cien)-36 b(ts)346 b([)p 0 .5 0 TeXcolorrgb 25076 61330 a SDict begin H.S end 25076 61330 a 0 .5 0 TeXcolorrgb Fs(5)p 0 .5 0 TeXcolorrgb 25727 60474 a SDict begin H.R end 25727 60474 a 25727 61330 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Merlini_MC) cvn H.B /ANN pdfmark end 25727 61330 a Black Fs(])g(to)g(pass)f(from)h(the)f (recurrences)g(to)h(the)f(corresp)36 b(ond-)800 62935 y(ing)466 b(generating)h(functions)f Fq(B)67 b Fs(\()p Fq(x)p Fs(\))466 b(and)f Fq(T)181 b Fs(\()p Fq(x)p Fs(\).)676 b(In)466 b(order)g(to)g(do)g(so,)475 b(w)-36 b(e)467 b(note)f(that)f(the)h(co)36 b(e\016cien)-36 b(ts)466 b(of)800 64540 y(the)433 b(sequence)h(generated)f(b)-36 b(y)433 b Fq(B)67 b Fs(\()p Fq(x)p Fs(\))434 b(b)36 b(egin:)14989 67200 y(1)p Fq(;)221 b Fs(1)297 b(+)d Fq(r)-36 b(;)221 b Fs(2)297 b(+)e(3)p Fq(r)331 b Fs(+)295 b Fq(r)25356 66651 y Fo(2)25882 67200 y Fq(;)221 b Fs(2)296 b(+)f(7)p Fq(r)332 b Fs(+)295 b(5)p Fq(r)32864 66651 y Fo(2)33685 67200 y Fs(+)g Fq(r)35614 66651 y Fo(3)36140 67200 y Fq(;)221 b(:)g(:)g(:)s(:)p Black 26150 70521 a Fs(12)p Black eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(Equation)434 b(\()p 0 .5 0 TeXcolorrgb 6924 -2672 a SDict begin H.S end 6924 -2672 a 0 .5 0 TeXcolorrgb Fs(7)p 0 .5 0 TeXcolorrgb 7574 -3528 a SDict begin H.R end 7574 -3528 a 7574 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.7) cvn H.B /ANN pdfmark end 7574 -2672 a Black Fs(\))f(b)36 b(ecomes)13687 -4277 y SDict begin H.S end 13687 -4277 a 13687 -4277 a SDict begin 14.5 H.A end 13687 -4277 a 13687 -4277 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark end 13687 -4277 a 11752 152 a Fq(B)67 b Fs(\()p Fq(x)p Fs(\))295 b Fn(\000)g Fs(1)h Fn(\000)f Fs(\(1)g(+)g Fq(r)36 b Fs(\))p Fq(x)296 b Fn(\000)f Fs(\(2)g(+)g(3)p Fq(r)332 b Fs(+)295 b Fq(r)30959 -330 y Fo(2)31485 152 y Fs(\))p Fq(x)32730 -330 y Fo(2)33551 152 y Fn(\000)g 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/Border [0 0 0] /Subtype /Link /Dest (equation.8) cvn H.B /ANN pdfmark end 10970 7646 a Black Fs(\))g(b)36 b(ecomes:)17589 6041 y SDict begin H.S end 17589 6041 a 17589 6041 a SDict begin 14.5 H.A end 17589 6041 a 17589 6041 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10) cvn H.B /DEST pdfmark end 17589 6041 a 3938 10470 a Fq(B)67 b Fs(\()p Fq(x)p Fs(\))295 b Fn(\000)g Fs(1)h Fn(\000)f Fs(\(1)g(+)g Fq(r)36 b Fs(\))p Fq(x)296 b Fn(\000)f Fs(\(2)g(+)g(3)p Fq(r)332 b Fs(+)295 b Fq(r)23145 9988 y Fo(2)23671 10470 y Fs(\))p Fq(x)24916 9988 y Fo(2)p 3938 11063 21504 54 v 14057 12280 a Fq(x)14796 11896 y Fo(3)25943 11369 y Fs(=)369 b(\()p Fq(r)331 b Fs(+)295 b(2\))31343 10470 y Fq(B)67 b Fs(\()p Fq(x)p Fs(\))295 b Fn(\000)g Fs(1)h Fn(\000)f Fs(\(1)h(+)e Fq(r)36 b Fs(\))p Fq(x)p 31343 11063 11329 54 v 36375 12280 a(x)37114 11896 y Fo(2)43100 11369 y Fn(\000)295 b Fq(T)181 b Fs(\()p Fq(x)p Fs(\))p Fq(:)3005 b Fs(\(10\))800 14715 y(W)-108 b(e)489 b(no)-36 b(w)490 b(ha)-36 b(v)g(e)490 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Fs(\()p Fq(x)p Fs(\))369 b(=)25897 25259 y(\()p Fq(r)332 b Fs(+)294 b(2\))i Fn(\000)f Fq(x)p 20751 25852 16541 54 v 20751 27069 a Fs(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(1\))p Fq(x)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(1\))p Fq(x)33899 26685 y Fo(2)34720 27069 y Fs(+)g Fq(x)36766 26685 y Fo(3)800 29227 y Fh(is)465 b(the)f(gener)-66 b(ating)463 b(function)g(of)h(the)h(pr)-66 b(o)g(duct)464 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))p Fq(a)p Fs(\()p Fq(n)296 b Fs(+)f(3;)221 b Fq(r)36 b Fs(\))p Fh(.)2751 31939 y Fs(The)434 b(author)f(is)h(indebted)e(to)h(an)h(anon) -36 b(ymous)433 b(review)-36 b(er)435 b(for)f(the)f(form)h(of)g(this)g (pro)36 b(of.)800 35150 y(W)-108 b(e)434 b(note)f(that)g(this)g (theorem)h(establishes)f(the)g(binomial)h(iden)-36 b(tit)g(y)35188 33545 y SDict begin H.S end 35188 33545 a 35188 33545 a SDict begin 14.5 H.A end 35188 33545 a 35188 33545 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11) cvn H.B /DEST pdfmark end 35188 33545 a 10986 37595 a Fp(k)10284 37993 y Fm(X)10426 40793 y Fp(j)51 b Fo(=0)12424 37382 y Fm(\022)13402 38262 y Fn(b)14125 37678 y Fp(n)p Fo(+1+2)p Fp(j)p 14125 37956 3408 54 v 15593 38720 a Fo(2)17665 38262 y Fn(c)14003 40261 y(b)14726 39677 y Fp(n)p Fl(\000)p Fo(2)p Fp(j)p 14726 39956 2205 54 v 15593 40719 a Fo(2)17064 40261 y Fn(c)18255 37382 y Fm(\023\022)20211 38198 y Fn(b)20934 37565 y Fp(n)p Fo(+2+2\()p Fp(k)24 b Fl(\000)p Fp(j)51 b Fo(\))p 20934 37892 5385 54 v 23391 38656 a(2)26451 38198 y Fn(c)20211 40325 y(b)20934 39692 y Fp(n)p Fo(+1)p Fl(\000)p Fo(2\()p Fp(k)24 b Fl(\000)p Fp(j)51 b Fo(\))p 20934 40020 V 23391 40783 a(2)26451 40325 y Fn(c)27042 37382 y Fm(\023)28388 39255 y Fs(=)30443 37595 y Fp(n)29769 37993 y Fm(X)29911 40793 y Fp(j)g Fo(=0)31688 39255 y Fs(\()p Fn(\000)p Fs(1\))34383 38707 y Fp(n)p Fl(\000)p Fp(j)36172 37382 y Fm(\022)37150 38357 y Fq(j)370 b Fs(+)295 b Fq(k)340 b Fs(+)295 b(1)37933 40166 y(2)p Fq(k)340 b Fs(+)295 b(1)42338 37382 y Fm(\023)50488 39255 y Fs(\(11\))800 43459 y(where)515 b(the)f(left)h(hand)f(side)g(follo)-36 b(ws)517 b(from)e(the)f(Cauc)-36 b(h)g(y)515 b(form)g(of)g(the)g(pro)36 b(duct)513 b(of)j(t)-36 b(w)g(o)514 b(p)36 b(olynomials)800 45064 y(and)452 b(the)g(righ)-36 b(t)452 b(hand)f(side)i(is)f(an)g (expression)h(for)g(the)f(general)h(term)f(of)h(the)f Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))453 b(co)36 b(e\016cien)-36 b(t)452 b(arra)-36 b(y)800 45560 y Fm(\020)4185 46512 y Fo(1)p 1726 46729 5389 54 v 1726 47493 a(\(1)p Fl(\000)p Fp(x)p Fo(\)\(1)p Fl(\000)p Fp(x)6288 47241 y Fk(2)6749 47493 y Fo(\))7248 47035 y Fq(;)9161 46512 y Fp(x)p 7963 46729 2926 54 v 7963 47493 a Fo(\(1)p Fl(\000)p Fp(x)p Fo(\))10427 47241 y Fk(2)11021 45560 y Fm(\021)11815 47035 y Fs(.)800 49900 y SDict begin H.S end 800 49900 a 800 49900 a SDict begin 14.5 H.A end 800 49900 a 800 49900 a SDict begin [ /View [/XYZ H.V] /Dest (section.4) cvn H.B /DEST pdfmark end 800 49900 a 1572 x Ft(4)2152 b Fe(a)p Fd(\()p Fe(n)p Fd(;)319 b Fe(r)52 b Fd(\))718 b 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a(2)19639 62329 y Fn(p)p 20747 62329 623 54 v 20747 63285 a Fq(r)22303 62368 y(U)23186 62567 y Fp(n)24034 60495 y Fm(\022)25012 62368 y Fn(\000)p Fq(i)26622 60513 y Fn(p)p 27728 60513 V 27728 61470 a Fq(r)p 26622 62063 1730 54 v 27161 63280 a Fs(2)28483 60495 y Fm(\023)29756 62368 y Fs(+)31196 60513 y Fn(p)p 32303 60513 623 54 v 957 x Fq(r)331 b Fn(\000)296 b Fs(1)p 31196 62063 4004 54 v 32008 63285 a(2)32658 62329 y Fn(p)p 33765 62329 623 54 v 956 x Fq(r)35332 62368 y(U)36215 62567 y Fp(n)37063 60495 y Fm(\022)38041 62368 y Fq(i)38618 60513 y Fn(p)p 39724 60513 V 39724 61470 a Fq(r)p 38618 62063 1730 54 v 39157 63280 a Fs(2)40479 60495 y Fm(\023)222 b(\033)42675 62368 y Fq(:)7452 b Fs(\(12\))p Black 26150 70521 a(13)p Black eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 1328 x Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(recall)g(that)f(the)g(g.f.)580 b(of)434 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))434 b(is)26441 -3195 y Fo(1+)p Fp(x)p Fl(\000)p Fp(x)29435 -3507 y Fk(2)p 25018 -2977 6301 54 v 25018 -2214 a Fo(1)p Fl(\000)p Fo(\()p Fp(r)26 b Fo(+2\))p Fp(x)29135 -2466 y Fk(2)29596 -2214 y Fo(+)p Fp(x)30858 -2466 y Fk(4)31452 -2672 y Fs(.)578 b(No)-36 b(w)3139 500 y(1)295 b(+)g Fq(x)h Fn(\000)f Fq(x)8493 18 y Fo(2)p 933 1093 10292 54 v 933 2309 a Fs(1)g Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)7832 1926 y Fo(2)8653 2309 y Fs(+)g Fq(x)10699 1926 y Fo(4)12465 1398 y Fs(=)18093 500 y(1)g Fn(\000)h Fq(x)21106 18 y Fo(2)p 14716 1093 V 14716 2309 a Fs(1)g Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)21615 1926 y Fo(2)22436 2309 y Fs(+)g Fq(x)24482 1926 y Fo(4)25436 1398 y Fs(+)31652 500 y Fq(x)p 26876 1093 V 26876 2309 a Fs(1)g Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)33775 1926 y Fo(2)34596 2309 y Fs(+)g Fq(x)36642 1926 y Fo(4)12465 5010 y Fs(=)14716 4112 y(1)p 14716 4705 651 54 v 14716 5922 a(2)15499 3137 y Fm(\032)20119 4112 y Fs(1)p 16628 4705 7631 54 v 16628 5927 a(1)h Fn(\000)18902 4971 y(p)p 20009 4971 623 54 v 956 x Fq(r)36 b(x)296 b Fn(\000)f Fq(x)23733 5544 y Fo(2)24687 5010 y Fs(+)29606 4112 y(1)p 26127 4705 7610 54 v 26127 5927 a(1)g(+)28379 4971 y Fn(p)p 29486 4971 623 54 v 956 x Fq(r)36 b(x)296 b Fn(\000)f Fq(x)33210 5544 y Fo(2)33869 3137 y Fm(\033)35160 5010 y Fs(+)37464 4112 y(1)p 36600 4705 2380 54 v 36600 5927 a(2)37250 4971 y Fn(p)p 38357 4971 623 54 v 956 x Fq(r)39112 3137 y Fm(\032)43731 4112 y Fs(1)p 40241 4705 7631 54 v 40241 5927 a(1)h Fn(\000)42515 4971 y(p)p 43622 4971 623 54 v 956 x Fq(r)36 b(x)296 b Fn(\000)f Fq(x)47346 5544 y Fo(2)48300 5010 y Fn(\000)53241 4112 y Fs(1)p 49761 4705 7610 54 v 49761 5927 a(1)h(+)52014 4971 y Fn(p)p 53120 4971 623 54 v 53120 5927 a Fq(r)37 b(x)295 b Fn(\000)g Fq(x)56844 5544 y Fo(2)57503 3137 y Fm(\033)12465 8641 y Fs(=)14583 6768 y Fm(\022)15694 7743 y Fs(1)p 15694 8336 651 54 v 15694 9552 a(2)16772 8641 y(+)19077 7743 y(1)p 18212 8336 2380 54 v 18212 9558 a(2)18862 8601 y Fn(p)p 19969 8601 623 54 v 957 x Fq(r)20724 6768 y Fm(\023)25547 7743 y Fs(1)p 22056 8336 7631 54 v 22056 9558 a(1)h Fn(\000)24330 8601 y(p)p 25437 8601 623 54 v 957 x Fq(r)36 b(x)296 b Fn(\000)f Fq(x)29161 9174 y Fo(2)30115 8641 y Fs(+)31422 6768 y Fm(\022)32533 7743 y Fs(1)p 32533 8336 651 54 v 32533 9552 a(2)33611 8641 y Fn(\000)35937 7743 y Fs(1)p 35072 8336 2380 54 v 35072 9558 a(2)35722 8601 y Fn(p)p 36829 8601 623 54 v 957 x Fq(r)37584 6768 y Fm(\023)42396 7743 y Fs(1)p 38916 8336 7610 54 v 38916 9558 a(1)h(+)41169 8601 y Fn(p)p 42276 8601 623 54 v 957 x Fq(r)36 b(x)295 b Fn(\000)h Fq(x)46000 9174 y Fo(2)12465 12308 y Fs(=)14716 10452 y Fn(p)p 15823 10452 V 957 x Fq(r)331 b Fs(+)295 b(1)p 14716 12002 3982 54 v 15517 13225 a(2)16167 12268 y Fn(p)p 17274 12268 623 54 v 957 x Fq(r)22453 11409 y Fs(1)p 18963 12002 7631 54 v 18963 13225 a(1)h Fn(\000)21237 12268 y(p)p 22344 12268 623 54 v 957 x Fq(r)36 b(x)296 b Fn(\000)f Fq(x)26068 12841 y Fo(2)27022 12308 y Fs(+)28462 10452 y Fn(p)p 29569 10452 V 957 x Fq(r)331 b Fn(\000)295 b Fs(1)p 28462 12002 4004 54 v 29273 13225 a(2)29923 12268 y Fn(p)p 31031 12268 623 54 v 31031 13225 a Fq(r)36210 11409 y Fs(1)p 32730 12002 7610 54 v 32730 13225 a(1)h(+)34983 12268 y Fn(p)p 36090 12268 623 54 v 957 x Fq(r)36 b(x)295 b Fn(\000)h Fq(x)39814 12841 y Fo(2)40472 12308 y Fq(:)800 15978 y Fs(The)434 b(assertion)g(follo)-36 b(ws)435 b(from)f(this,)g (giv)-36 b(en)434 b(Eq.)579 b(\()p 0 .5 0 TeXcolorrgb 25560 15978 a SDict begin H.S end 25560 15978 a 0 .5 0 TeXcolorrgb Fs(2)p 0 .5 0 TeXcolorrgb 26210 15122 a SDict begin H.R end 26210 15122 a 26210 15978 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.2) cvn H.B /ANN pdfmark end 26210 15978 a Black Fs(\).)p 51860 15978 45 878 v 51905 15145 781 45 v 51905 15978 V 52684 15978 45 878 v 800 17307 a SDict begin H.S end 800 17307 a 800 17307 a SDict begin 14.5 H.A end 800 17307 a 800 17307 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8) cvn H.B /DEST pdfmark end 800 17307 a 800 17418 a SDict begin H.S end 800 17418 a 800 17418 a SDict begin 14.5 H.A end 800 17418 a 800 17418 a SDict begin [ /View [/XYZ H.V] /Dest (equation.13) cvn H.B /DEST pdfmark end 800 17418 a Black 1272 x Fg(Corollary)521 b(8.)p Black 12482 21021 a Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)f Fq(i)18352 20473 y Fp(n)18978 21021 y Fq(U)19861 21220 y Fp(n)20709 19148 y Fm(\022)21687 21021 y Fq(i)22264 19166 y Fn(p)p 23370 19166 623 54 v 23370 20122 a Fq(r)p 22264 20716 1730 54 v 22803 21932 a Fs(2)24125 19148 y Fm(\023)221 b(\032)26321 21021 y Fs(\()p Fn(\000)p Fs(1\))29016 20473 y Fp(n)29775 19166 y Fn(p)p 30882 19166 623 54 v 956 x Fq(r)331 b Fs(+)295 b(1)p 29775 20716 3982 54 v 30576 21938 a(2)31226 20981 y Fn(p)p 32333 20981 623 54 v 957 x Fq(r)34184 21021 y Fs(+)35624 19166 y Fn(p)p 36731 19166 V 956 x Fq(r)332 b Fn(\000)295 b Fs(1)p 35624 20716 4004 54 v 36436 21938 a(2)37086 20981 y Fn(p)p 38193 20981 623 54 v 957 x Fq(r)39760 19148 y Fm(\033)40756 21021 y Fq(:)9371 b Fs(\(13\))p Black 800 24497 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(ha)-36 b(v)g(e)433 b Fq(U)10993 24696 y Fp(n)11620 24497 y Fs(\()p Fn(\000)p Fq(x)p Fs(\))369 b(=)g(\()p Fn(\000)p Fs(1\))18849 24015 y Fp(n)19475 24497 y Fq(U)20358 24696 y Fp(n)20985 24497 y Fs(\()p Fq(x)p Fs(\).)578 b(The)433 b(result)g(follo)-36 b(ws)436 b(from)e(this.)p 51860 24497 45 878 v 51905 23663 781 45 v 51905 24497 V 52684 24497 45 878 v 800 27098 a(This)592 b(no)-36 b(w)592 b(allo)-36 b(ws)593 b(us)e(to)h(c)-36 b(haracterize)592 b(the)f(pro)36 b(duct)590 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))p Fq(a)p Fs(\()p Fq(n)404 b Fs(+)e(1;)221 b Fq(r)36 b Fs(\))593 b(in)e(terms)h(of)g(Cheb)-36 b(yshev)800 28703 y(p)36 b(olynomials.)800 28962 y SDict begin H.S end 800 28962 a 800 28962 a SDict begin 14.5 H.A end 800 28962 a 800 28962 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9) cvn H.B /DEST pdfmark end 800 28962 a 800 30069 a SDict begin H.S end 800 30069 a 800 30069 a SDict begin 14.5 H.A end 800 30069 a 800 30069 a SDict begin [ /View [/XYZ H.V] /Dest (equation.14) cvn H.B /DEST pdfmark end 800 30069 a Black 1346 x Fg(Prop)42 b(osition)500 b(9.)p Black 12352 36022 a Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))p Fq(a)p Fs(\()p Fq(n)296 b Fs(+)f(1;)221 b Fq(r)36 b Fs(\))370 b(=)f Fq(i)p Fs(\()p Fn(\000)p Fs(1\))26845 35473 y Fp(n)p Fo(+1)28806 34640 y Fq(U)29689 34839 y Fp(n)30537 33166 y Fm(\020)31330 34640 y Fq(i)31907 33383 y Fl(p)p 32691 33383 451 40 v 638 x Fp(r)p 31907 34335 1235 54 v 32289 35098 a Fo(2)33275 33166 y Fm(\021)34289 34640 y Fq(U)35172 34839 y Fp(n)p Fo(+1)37223 33166 y Fm(\020)38016 34640 y Fq(i)38593 33383 y Fl(p)p 39377 33383 451 40 v 638 x Fp(r)p 38593 34335 1235 54 v 38975 35098 a Fo(2)39960 33166 y Fm(\021)p 28806 35716 11948 54 v 33915 35982 a Fn(p)p 35022 35982 623 54 v 957 x Fq(r)40886 36022 y(:)9241 b Fs(\(14\))p Black 800 39619 a Fh(Pr)-66 b(o)g(of.)p Black 3359 43278 a Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))p Fq(a)p Fs(\()p Fq(n)296 b Fs(+)f(1;)221 b Fq(r)36 b Fs(\))1108 b(=)f Fq(i)p Fs(\()p Fn(\000)p Fs(1\))19328 42729 y Fp(n)19954 43278 y Fq(U)20837 43477 y Fp(n)21685 41405 y Fm(\022)22663 43278 y Fq(i)23240 41422 y Fn(p)p 24346 41422 V 24346 42379 a Fq(r)p 23240 42972 1730 54 v 23779 44189 a Fs(2)25101 41405 y Fm(\023)26301 43278 y Fq(U)27184 43477 y Fp(n)p Fo(+1)29234 41405 y Fm(\022)30212 43278 y Fq(i)30789 41422 y Fn(p)p 31895 41422 623 54 v 31895 42379 a Fq(r)p 30789 42972 1730 54 v 31328 44189 a Fs(2)32650 41405 y Fm(\023)221 b(\032)34846 43278 y Fs(\()p Fn(\000)p Fs(1\))37541 42729 y Fp(n)38300 41422 y Fn(p)p 39407 41422 623 54 v 957 x Fq(r)331 b Fs(+)295 b(1)p 38300 42972 3982 54 v 39101 44195 a(2)39751 43238 y Fn(p)p 40858 43238 623 54 v 957 x Fq(r)42710 43278 y Fs(+)44149 41422 y Fn(p)p 45256 41422 V 957 x Fq(r)332 b Fn(\000)295 b Fs(1)p 44149 42972 4004 54 v 44961 44195 a(2)45611 43238 y Fn(p)p 46718 43238 623 54 v 957 x Fq(r)48285 41405 y Fm(\033)49577 43278 y Fn(\001)17489 45071 y Fm(\032)18781 46944 y Fn(\000)g Fs(\()p Fn(\000)p Fs(1\))22804 46396 y Fp(n)23563 45089 y Fn(p)p 24670 45089 V 957 x Fq(r)332 b Fs(+)295 b(1)p 23563 46639 3982 54 v 24364 47861 a(2)25014 46904 y Fn(p)p 26122 46904 623 54 v 26122 47861 a Fq(r)27973 46944 y Fs(+)29413 45089 y Fn(p)p 30520 45089 V 957 x Fq(r)331 b Fn(\000)295 b Fs(1)p 29413 46639 4004 54 v 30224 47861 a(2)30874 46904 y Fn(p)p 31982 46904 623 54 v 31982 47861 a Fq(r)33549 45071 y Fm(\033)14070 50611 y Fs(=)1107 b Fq(i)p Fs(\()p Fn(\000)p Fs(1\))19328 50062 y Fp(n)19954 50611 y Fq(U)20837 50810 y Fp(n)21685 48738 y Fm(\022)22663 50611 y Fq(i)23240 48755 y Fn(p)p 24346 48755 V 24346 49712 a Fq(r)p 23240 50305 1730 54 v 23779 51522 a Fs(2)25101 48738 y Fm(\023)26301 50611 y Fq(U)27184 50810 y Fp(n)p Fo(+1)29234 48738 y Fm(\022)30212 50611 y Fq(i)30789 48755 y Fn(p)p 31895 48755 623 54 v 31895 49712 a Fq(r)p 30789 50305 1730 54 v 31328 51522 a Fs(2)32650 48738 y Fm(\023)34293 49712 y Fs(1)p 33982 50305 1273 54 v 33982 51522 a(4)p Fq(r)35388 50611 y Fs(\()p Fn(\000)p Fs(\()37433 49589 y Fn(p)p 38539 49589 623 54 v 38539 50611 a Fq(r)332 b Fs(+)295 b(1\))41920 50062 y Fo(2)42741 50611 y Fs(+)g(\()44554 49589 y Fn(p)p 45661 49589 V 1022 x Fq(r)331 b Fn(\000)295 b Fs(1\))49062 50062 y Fo(2)49588 50611 y Fs(\))14070 54277 y(=)1107 b Fq(i)p Fs(\()p Fn(\000)p Fs(1\))19328 53729 y Fp(n)19954 54277 y Fq(U)20837 54476 y Fp(n)21685 52404 y Fm(\022)22663 54277 y Fq(i)23240 52422 y Fn(p)p 24346 52422 V 24346 53378 a Fq(r)p 23240 53972 1730 54 v 23779 55188 a Fs(2)25101 52404 y Fm(\023)26301 54277 y Fq(U)27184 54476 y Fp(n)p Fo(+1)29234 52404 y Fm(\022)30212 54277 y Fq(i)30789 52422 y Fn(p)p 31895 52422 623 54 v 31895 53378 a Fq(r)p 30789 53972 1730 54 v 31328 55188 a Fs(2)32650 52404 y Fm(\023)34293 53378 y Fs(1)p 33982 53972 1273 54 v 33982 55188 a(4)p Fq(r)35388 54277 y(:)295 b Fn(\000)h Fs(4)38023 53255 y Fn(p)p 39130 53255 623 54 v 1022 x Fq(r)14070 58891 y Fs(=)1107 b Fq(i)p Fs(\()p Fn(\000)p Fs(1\))19328 58342 y Fp(n)p Fo(+1)21289 57509 y Fq(U)22172 57708 y Fp(n)23020 56035 y Fm(\020)23813 57509 y Fq(i)24390 56252 y Fl(p)p 25174 56252 451 40 v 638 x Fp(r)p 24390 57204 1235 54 v 24772 57967 a Fo(2)25758 56035 y Fm(\021)26772 57509 y Fq(U)27655 57708 y Fp(n)p Fo(+1)29706 56035 y Fm(\020)30499 57509 y Fq(i)31076 56252 y Fl(p)p 31860 56252 451 40 v 638 x Fp(r)p 31076 57204 1235 54 v 31458 57967 a Fo(2)32443 56035 y Fm(\021)p 21289 58585 11948 54 v 26398 58851 a Fn(p)p 27505 58851 623 54 v 957 x Fq(r)33369 58891 y(:)p 51860 62443 45 878 v 51905 61610 781 45 v 51905 62443 V 52684 62443 45 878 v 800 65045 a Fs(The)578 b(follo)-36 b(wing)580 b(prop)36 b(osition)578 b(pro)-36 b(vides)578 b(a)g(link)g(b)36 b(et)-36 b(w)g(een)578 b(the)f(p)36 b(olynomials)579 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))579 b(and)e(con)-36 b(tin)g(ued)800 66650 y(fractions.)800 66650 y SDict begin H.S end 800 66650 a 800 66650 a SDict begin 14.5 H.A end 800 66650 a 800 66650 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.10) cvn H.B /DEST pdfmark end 800 66650 a Black 26150 70521 a Fs(14)p Black eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.15) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (equation.15) cvn H.B /DEST pdfmark end 800 -4000 a Black 1328 x Fg(Prop)42 b(osition)500 b(10.)p Black 10023 572 a Fq(a)p Fs(\()p Fq(n)295 b Fs(+)g(2;)221 b Fq(r)36 b Fs(\))370 b(=)f Fq(a)p Fs(\()p Fq(n)295 b Fs(+)g(1;)221 b Fq(r)36 b Fs(\))23761 -327 y Fq(r)g Fs(\(1)296 b(+)f(\()p Fn(\000)p Fs(1\))29837 -809 y Fp(n)30463 -327 y Fs(\))g(+)g(1)h Fn(\000)f Fs(\()p Fn(\000)p Fs(1\))37540 -809 y Fp(n)p 23761 266 14405 54 v 30639 1483 a Fs(2)38595 572 y(+)f Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))6912 b(\(15\))p Black 800 3495 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(ha)-36 b(v)g(e)2573 6805 y Fq(a)p Fs(\()p Fq(n)295 b Fs(+)g(2;)221 b Fq(r)36 b Fs(\))1108 b(=)e Fq(i)12170 6257 y Fp(n)p Fo(+2)13999 6805 y Fq(U)14882 7004 y Fp(n)p Fo(+2)16932 4932 y Fm(\022)18043 5907 y Fq(i)18487 4950 y Fn(p)p 19593 4950 623 54 v 19593 5907 a Fq(r)p 18043 6500 2173 54 v 18804 7716 a Fs(2)20348 4932 y Fm(\023)222 b(\032)22544 6805 y Fs(\()p Fn(\000)p Fs(1\))25239 6257 y Fp(n)p Fo(+2)27200 4950 y Fn(p)p 28307 4950 623 54 v 957 x Fq(r)332 b Fs(+)295 b(1)p 27200 6500 3982 54 v 28001 7722 a(2)28651 6765 y Fn(p)p 29759 6765 623 54 v 29759 7722 a Fq(r)31610 6805 y Fs(+)33050 4950 y Fn(p)p 34157 4950 V 957 x Fq(r)331 b Fn(\000)295 b Fs(1)p 33050 6500 4004 54 v 33861 7722 a(2)34511 6765 y Fn(p)p 35619 6765 623 54 v 35619 7722 a Fq(r)37186 4932 y Fm(\033)9608 10472 y Fs(=)1106 b Fn(\000)p Fq(i)13203 9923 y Fp(n)14051 8599 y Fm(\022)15029 10472 y Fs(2)p Fq(i)16256 8616 y Fn(p)p 17363 8616 V 957 x Fq(r)p 16256 10166 1730 54 v 16795 11383 a Fs(2)18118 10472 y Fq(U)19001 10671 y Fp(n)p Fo(+1)21051 8599 y Fm(\022)22162 9573 y Fq(i)22606 8616 y Fn(p)p 23712 8616 623 54 v 23712 9573 a Fq(r)p 22162 10166 2173 54 v 22923 11383 a Fs(2)24467 8599 y Fm(\023)25740 10472 y Fn(\000)296 b Fq(U)27952 10671 y Fp(n)28800 8599 y Fm(\022)29910 9573 y Fq(i)30354 8616 y Fn(p)p 31461 8616 623 54 v 957 x Fq(r)p 29910 10166 2173 54 v 30672 11383 a Fs(2)32216 8599 y Fm(\023\023)221 b(\032)35389 10472 y Fs(\()p Fn(\000)p Fs(1\))38084 9923 y Fp(n)p Fo(+2)40046 8616 y Fn(p)p 41153 8616 623 54 v 957 x Fq(r)331 b Fs(+)295 b(1)p 40046 10166 3982 54 v 40847 11388 a(2)41497 10432 y Fn(p)p 42604 10432 623 54 v 956 x Fq(r)44456 10472 y Fs(+)45895 8616 y Fn(p)p 47002 8616 V 957 x Fq(r)332 b Fn(\000)295 b Fs(1)p 45895 10166 4004 54 v 46707 11388 a(2)47357 10432 y Fn(p)p 48464 10432 623 54 v 956 x Fq(r)50031 8599 y Fm(\033)9608 14138 y Fs(=)1106 b Fn(\000)p Fq(i)13203 13590 y Fp(n)p Fo(+1)15032 13116 y Fn(p)p 16139 13116 V 1022 x Fq(r)36 b(U)17644 14337 y Fp(n)p Fo(+1)19694 12265 y Fm(\022)20805 13239 y Fq(i)21249 12283 y Fn(p)p 22356 12283 V 956 x Fq(r)p 20805 13832 2173 54 v 21566 15049 a Fs(2)23111 12265 y Fm(\023)221 b(\032)25306 14138 y Fs(\()p Fn(\000)p Fs(1\))28001 13590 y Fp(n)28760 12283 y Fn(p)p 29867 12283 623 54 v 956 x Fq(r)332 b Fs(+)295 b(1)p 28760 13832 3982 54 v 29561 15055 a(2)30211 14098 y Fn(p)p 31319 14098 623 54 v 31319 15055 a Fq(r)33170 14138 y Fs(+)34610 12283 y Fn(p)p 35717 12283 V 956 x Fq(r)331 b Fn(\000)295 b Fs(1)p 34610 13832 4004 54 v 35421 15055 a(2)36071 14098 y Fn(p)p 37179 14098 623 54 v 37179 15055 a Fq(r)38746 12265 y Fm(\033)40037 14138 y Fs(+)16929 17804 y Fq(i)17373 17256 y Fp(n)17999 17804 y Fq(U)18882 18003 y Fp(n)19730 15931 y Fm(\022)20841 16906 y Fq(i)21285 15949 y Fn(p)p 22391 15949 V 22391 16906 a Fq(r)p 20841 17499 2173 54 v 21602 18716 a Fs(2)23146 15931 y Fm(\023)222 b(\032)25342 17804 y Fs(\()p Fn(\000)p Fs(1\))28037 17256 y Fp(n)28796 15949 y Fn(p)p 29903 15949 623 54 v 957 x Fq(r)332 b Fs(+)294 b(1)p 28796 17499 3982 54 v 29597 18721 a(2)30247 17765 y Fn(p)p 31354 17765 623 54 v 956 x Fq(r)33206 17804 y Fs(+)34645 15949 y Fn(p)p 35752 15949 V 957 x Fq(r)332 b Fn(\000)295 b Fs(1)p 34645 17499 4004 54 v 35457 18721 a(2)36107 17765 y Fn(p)p 37214 17765 623 54 v 956 x Fq(r)38781 15931 y Fm(\033)9608 23215 y Fs(=)1106 b Fq(a)p Fs(\()p Fq(n)295 b Fs(+)g(2;)221 b Fq(r)36 b Fs(\))17786 21435 y Fn(\000)18819 20478 y(p)p 19927 20478 V 19927 21435 a Fq(r)20550 19562 y Fm(\032)21546 21435 y Fs(\()p Fn(\000)p Fs(1\))24241 20953 y Fp(n)25000 20178 y Fl(p)p 25784 20178 451 40 v 638 x Fp(r)26 b Fo(+1)p 25000 21130 2438 54 v 25366 21901 a(2)25836 21263 y Fl(p)p 26621 21263 451 40 v 26621 21901 a Fp(r)27865 21435 y Fs(+)29305 20178 y Fl(p)p 30089 20178 V 638 x Fp(r)g Fl(\000)p Fo(1)p 29305 21130 2438 54 v 29671 21901 a(2)30141 21263 y Fl(p)p 30925 21263 451 40 v 638 x Fp(r)31875 19562 y Fm(\033)p 17786 22910 15084 54 v 18567 23122 a(\032)19564 24995 y Fs(\()p Fn(\000)p Fs(1\))22259 24612 y Fp(n)p Fo(+1)24220 23738 y Fl(p)p 25004 23738 451 40 v 638 x Fp(r)g Fo(+1)p 24220 24690 2438 54 v 24586 25461 a(2)25056 24823 y Fl(p)p 25841 24823 451 40 v 25841 25461 a Fp(r)27085 24995 y Fs(+)28525 23738 y Fl(p)p 29309 23738 V 638 x Fp(r)g Fl(\000)p Fo(1)p 28525 24690 2438 54 v 28891 25461 a(2)29361 24823 y Fl(p)p 30145 24823 451 40 v 638 x Fp(r)31095 23122 y Fm(\033)33299 23215 y Fs(+)295 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))9608 28595 y(=)1106 b Fq(a)p Fs(\()p Fq(n)295 b Fs(+)g(1;)221 b Fq(r)36 b Fs(\))17786 27696 y Fq(r)g Fs(\(1)297 b(+)e(\()p Fn(\000)p Fs(1\))23863 27214 y Fp(n)24489 27696 y Fs(\))g(+)g(1)g Fn(\000)h Fs(\()p Fn(\000)p Fs(1\))31566 27214 y Fp(n)p 17786 28289 14405 54 v 24665 29506 a Fs(2)32620 28595 y(+)f Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))p Fq(:)p 51860 31474 45 878 v 51905 30640 781 45 v 51905 31474 V 52684 31474 45 878 v 800 33976 a Fs(Inspired)433 b(b)-36 b(y)433 b([)p 0 .5 0 TeXcolorrgb 7996 33976 a SDict begin H.S end 7996 33976 a 0 .5 0 TeXcolorrgb Fs(4)p 0 .5 0 TeXcolorrgb 8647 33120 a SDict begin H.R end 8647 33120 a 8647 33976 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Mansour) cvn H.B /ANN pdfmark end 8647 33976 a Black Fs(],)h(w)-36 b(e)434 b(no)-36 b(w)434 b(de\014ne)e(a)i(family)h(of)g(rational)f (functions)f Fq(R)11 b Fs(\()p Fq(n;)221 b(r)36 b Fs(\))435 b(as)f(follo)-36 b(ws)217 b(:)16944 38339 y Fq(R)11 b Fs(\()p Fq(n;)221 b(r)36 b Fs(\))370 b(=)22688 35616 y Fm(8)22688 36812 y(<)22688 39203 y(:)24422 36521 y Fs(1)6529 b(if)434 b Fq(n)369 b Fs(=)g(0)24422 38126 y(1)6529 b(if)434 b Fq(n)369 b Fs(=)g(1)27223 39208 y Fo(1)p 24555 39426 5806 54 v 24555 40269 a(1+)25890 39907 y Fk(1)p 25890 40068 407 40 v 25890 40616 a Fj(r)26430 40269 y Fp(R)5 b Fo(\()p Fp(n)p Fl(\000)p Fo(1)p Fp(;r)26 b Fo(\))31601 39731 y Fs(if)434 b Fq(n)369 b(>)g Fs(1)800 42629 y(The)434 b(\014rst)e(elemen)-36 b(ts)434 b(of)g(the)f(family)i (are)13957 45929 y(1)p Fq(;)221 b Fs(1)p Fq(;)17682 45030 y(r)p 16556 45623 2875 54 v 16556 46840 a(r)332 b Fs(+)295 b(1)19564 45929 y Fq(;)20279 45030 y(r)332 b Fs(+)294 b(1)p 20279 45623 V 20279 46840 a Fq(r)332 b Fs(+)294 b(2)23287 45929 y Fq(;)24885 45030 y(r)36 b Fs(\()p Fq(r)332 b Fs(+)295 b(2\))p 24002 45623 6276 54 v 24002 46840 a Fq(r)24624 46456 y Fo(2)25445 46840 y Fs(+)g(3)p Fq(r)332 b Fs(+)295 b(1)30410 45929 y Fq(;)31126 45030 y(r)31748 44548 y Fo(2)32569 45030 y Fs(+)g(3)p Fq(r)331 b Fs(+)295 b(1)p 31126 45623 V 31126 46840 a Fq(r)31748 46456 y Fo(2)32569 46840 y Fs(+)g(4)p Fq(r)331 b Fs(+)295 b(3)37533 45929 y Fq(;)221 b(:)g(:)g(:)800 47921 y SDict begin H.S end 800 47921 a 800 47921 a SDict begin 14.5 H.A end 800 47921 a 800 47921 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11) cvn H.B /DEST pdfmark end 800 47921 a Black 1033 x Fg(Prop)42 b(osition)500 b(11.)p Black 17848 50559 a Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)e Fh(Denominator)o Fs(\()p Fq(R)11 b Fs(\()p Fq(n;)221 b(r)36 b Fs(\)\))p Black 800 52972 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(Induction,)433 b(using)h(Eq.)578 b(\()p 0 .5 0 TeXcolorrgb 17542 52972 a SDict begin H.S end 17542 52972 a 0 .5 0 TeXcolorrgb Fs(15)p 0 .5 0 TeXcolorrgb 18843 52116 a SDict begin H.R end 18843 52116 a 18843 52972 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.15) cvn H.B /ANN pdfmark end 18843 52972 a Black Fs(\).)p 51860 52972 45 878 v 51905 52138 781 45 v 51905 52972 V 52684 52972 45 878 v 800 55473 a(Noting)434 b(that)16828 58751 y Fq(R)11 b Fs(\()p Fq(n;)221 b(r)36 b Fs(\))370 b(=)22704 57370 y Fq(i)23148 56413 y Fn(p)p 24255 56413 623 54 v 957 x Fq(r)36 b(U)25760 57569 y Fp(n)p Fl(\000)p Fo(1)27811 55895 y Fm(\020)28737 56751 y Fp(i)29057 56112 y Fl(p)p 29841 56112 451 40 v 639 x Fp(r)p 28737 57064 1556 54 v 29279 57828 a Fo(2)30425 55895 y Fm(\021)p 22704 58446 8514 54 v 24392 60133 a Fq(U)25275 60332 y Fp(n)26123 58658 y Fm(\020)27049 59514 y Fp(i)27369 58875 y Fl(p)p 28154 58875 451 40 v 28154 59514 a Fp(r)p 27049 59827 1556 54 v 27592 60591 a Fo(2)28737 58658 y Fm(\021)31351 58751 y Fq(;)1522 b(n)370 b(>)e Fs(0)p Fq(;)800 62599 y Fs(w)-36 b(e)434 b(ha)-36 b(v)g(e)5714 60994 y SDict begin H.S end 5714 60994 a 5714 60994 a SDict begin 14.5 H.A end 5714 60994 a 5714 60994 a SDict begin [ /View [/XYZ H.V] /Dest (equation.16) cvn H.B /DEST pdfmark end 5714 60994 a 14421 65619 a Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)e(Denominator)27492 62949 y Fm(0)27492 65340 y(@)28787 64238 y Fq(i)29231 63281 y Fn(p)p 30338 63281 623 54 v 957 x Fq(r)36 b(U)31843 64437 y Fp(n)p Fl(\000)p Fo(1)33894 62763 y Fm(\020)34820 63619 y Fp(i)35140 62980 y Fl(p)p 35924 62980 451 40 v 639 x Fp(r)p 34820 63932 1556 54 v 35362 64696 a Fo(2)36508 62763 y Fm(\021)p 28787 65314 8514 54 v 30475 67001 a Fq(U)31358 67200 y Fp(n)32206 65526 y Fm(\020)33132 66382 y Fp(i)33452 65743 y Fl(p)p 34237 65743 451 40 v 34237 66382 a Fp(r)p 33132 66695 1556 54 v 33675 67459 a Fo(2)34820 65526 y Fm(\021)37434 62949 y(1)37434 65340 y(A)38818 65619 y Fq(:)11309 b Fs(\(16\))p Black 26150 70521 a(15)p Black eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.16) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(Finally)-108 b(,)435 b(w)-36 b(e)433 b(can)h(use)f(the)g(expression)16812 1200 y Fq(U)17695 1399 y Fp(n)18322 1200 y Fs(\()p Fq(x)p Fs(\))368 b(=)h(2)22472 652 y Fp(n)23892 -460 y(n)23329 -62 y Fm(Y)23320 2767 y Fp(k)24 b Fo(=1)25257 -673 y Fm(\022)26235 1200 y Fq(x)295 b Fn(\000)h Fs(cos)30560 -673 y Fm(\022)32432 302 y Fq(k)45 b(\031)p 31671 895 3029 54 v 31671 2111 a(n)296 b Fs(+)e(1)34833 -673 y Fm(\023)o(\023)800 5335 y Fs(and)433 b(Corollary)p 0 .5 0 TeXcolorrgb 9092 5335 a SDict begin H.S end 9092 5335 a 0 .5 0 TeXcolorrgb Fg(13)p 0 .5 0 TeXcolorrgb 10586 4479 a SDict begin H.R end 10586 4479 a 10586 5335 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 10586 5335 a Black 434 w Fs(to)g(giv)-36 b(e)435 b(another)e(c)-36 b(haracterization)434 b(of)h Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\).)800 5667 y SDict begin H.S end 800 5667 a 800 5667 a SDict begin 14.5 H.A end 800 5667 a 800 5667 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.12) cvn H.B /DEST pdfmark end 800 5667 a 800 6774 a SDict begin H.S end 800 6774 a 800 6774 a SDict begin 14.5 H.A end 800 6774 a 800 6774 a SDict begin [ /View [/XYZ H.V] /Dest (equation.17) cvn H.B /DEST pdfmark end 800 6774 a Black 1273 x Fg(Prop)42 b(osition)500 b(12.)p Black 16432 11103 a Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)21909 9213 y Fl(b)22460 8851 y Fj(n)p 22460 9012 512 40 v 22513 9561 a Fk(2)23104 9213 y Fl(c)21867 9841 y Fm(Y)21858 12670 y Fp(k)24 b Fo(=1)23795 9230 y Fm(\022)24773 11103 y Fq(r)331 b Fs(+)295 b(4)221 b(cos)29610 10555 y Fo(2)30357 9230 y Fm(\022)32229 10204 y Fq(k)45 b(\031)p 31468 10798 3029 54 v 31468 12014 a(n)296 b Fs(+)e(1)34629 9230 y Fm(\023\023)36807 11103 y Fq(:)13320 b Fs(\(17\))p Black 800 14943 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(b)36 b(egin)433 b(with)h(the)f(case)h Fq(n)369 b Fs(=)g(2)p Fq(m)p Fs(.)578 b(Then)5928 18815 y Fq(i)6372 18266 y Fp(n)6998 18815 y Fq(U)7881 19014 y Fp(n)8729 16942 y Fm(\022)9840 17916 y Fq(i)10284 16959 y Fn(p)p 11390 16959 623 54 v 11390 17916 a Fq(r)p 9840 18509 2173 54 v 10601 19726 a Fs(2)12145 16942 y Fm(\023)14230 18815 y Fs(=)1107 b Fq(i)16793 18266 y Fp(n)17419 18815 y Fs(2)18069 18266 y Fp(n)19489 17154 y(n)18926 17553 y Fm(Y)18917 20382 y Fp(k)24 b Fo(=1)20854 16942 y Fm(\022)21965 17916 y Fq(i)22409 16959 y Fn(p)p 23515 16959 623 54 v 23515 17916 a Fq(r)p 21965 18509 2173 54 v 22726 19726 a Fs(2)24566 18815 y Fn(\000)295 b Fs(cos)27857 16942 y Fm(\022)29729 17916 y Fq(k)45 b(\031)p 28967 18509 3029 54 v 28967 19726 a(n)296 b Fs(+)f(1)32129 16942 y Fm(\023\023)14230 23132 y Fs(=)1107 b Fq(i)16793 22584 y Fp(n)18213 21472 y(n)17649 21870 y Fm(Y)17640 24699 y Fp(k)24 b Fo(=1)19577 21259 y Fm(\022)20555 23132 y Fq(i)20999 22110 y Fn(p)p 22106 22110 623 54 v 1022 x Fq(r)331 b Fn(\000)296 b Fs(2)221 b(cos)27186 21259 y Fm(\022)29058 22233 y Fq(k)45 b(\031)p 28297 22827 3029 54 v 28297 24043 a(n)295 b Fs(+)g(1)31458 21259 y Fm(\023\023)14230 27449 y Fs(=)1107 b Fq(i)16793 26901 y Fp(n)18082 25789 y(m)17649 26187 y Fm(Y)17640 29016 y Fp(k)24 b Fo(=1)19577 25576 y Fm(\022)20555 27449 y Fq(i)20999 26428 y Fn(p)p 22106 26428 623 54 v 1021 x Fq(r)331 b Fn(\000)296 b Fs(2)221 b(cos)27186 25576 y Fm(\022)29058 26551 y Fq(k)45 b(\031)p 28297 27144 3029 54 v 28297 28360 a(n)295 b Fs(+)g(1)31458 25576 y Fm(\023\023)221 b(\022)34613 27449 y Fq(i)35057 26428 y Fn(p)p 36164 26428 623 54 v 1021 x Fq(r)331 b Fs(+)295 b(2)221 b(cos)41223 25576 y Fm(\022)43095 26551 y Fq(k)45 b(\031)p 42333 27144 3029 54 v 42333 28360 a(n)296 b Fs(+)f(1)45495 25576 y Fm(\023\023)14230 31766 y Fs(=)1107 b Fq(i)16793 31218 y Fp(n)18082 30106 y(m)17649 30504 y Fm(Y)17640 33333 y Fp(k)24 b Fo(=1)19577 29893 y Fm(\022)20555 31766 y Fn(\000)p Fq(r)332 b Fn(\000)295 b Fs(4)221 b(cos)26447 31218 y Fo(2)27195 29893 y Fm(\022)29066 30868 y Fq(k)45 b(\031)p 28305 31461 V 28305 32678 a(n)296 b Fs(+)f(1)31467 29893 y Fm(\023\023)14230 36084 y Fs(=)1107 b Fq(i)16793 35535 y Fo(2)p Fp(m)18151 36084 y Fs(\()p Fn(\000)p Fs(1\))20846 35535 y Fp(m)22397 34423 y(m)21964 34822 y Fm(Y)21955 37651 y Fp(k)24 b Fo(=1)23892 34211 y Fm(\022)24870 36084 y Fq(r)331 b Fs(+)295 b(4)221 b(cos)29707 35535 y Fo(2)30454 34211 y Fm(\022)32326 35185 y Fq(k)45 b(\031)p 31565 35778 V 31565 36995 a(n)296 b Fs(+)e(1)34727 34211 y Fm(\023)o(\023)14230 40913 y Fs(=)16400 39023 y Fl(b)16951 38661 y Fj(n)p 16951 38822 512 40 v 17004 39370 a Fk(2)17595 39023 y Fl(c)16358 39651 y Fm(Y)16349 42480 y Fp(k)24 b Fo(=1)18286 39040 y Fm(\022)19264 40913 y Fq(r)331 b Fs(+)295 b(4)221 b(cos)24101 40364 y Fo(2)24848 39040 y Fm(\022)26720 40014 y Fq(k)45 b(\031)p 25959 40607 3029 54 v 25959 41824 a(n)296 b Fs(+)f(1)29121 39040 y Fm(\023\023)31298 40913 y Fq(:)800 44974 y Fs(Note)424 b(that)g(in)g(the)f(pro)36 b(duct)423 b(expansion)h(w)-36 b(e)425 b(ha)-36 b(v)g(e)424 b(used)f(a)h(\014rst)g(with)g(last,)i (second)e(with)g(next-to-last,)p Black 26150 70521 a(16)p Black eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.17) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(etc)434 b(coupling.)578 b(W)-108 b(e)434 b(no)-36 b(w)433 b(turn)g(to)g(the)g(case)h Fq(n)370 b Fs(=)e(2)p Fq(m)296 b Fs(+)f(1.)578 b(W)-108 b(e)434 b(obtain)4452 1200 y Fq(i)4896 652 y Fp(n)5522 1200 y Fq(U)6405 1399 y Fp(n)7253 -673 y Fm(\022)8364 302 y Fq(i)8808 -655 y Fn(p)p 9914 -655 623 54 v 9914 302 a Fq(r)p 8364 895 2173 54 v 9125 2111 a Fs(2)10670 -673 y Fm(\023)12754 1200 y Fs(=)1107 b Fq(i)15317 652 y Fp(n)15943 1200 y Fs(2)16593 652 y Fp(n)18013 -460 y(n)17450 -62 y Fm(Y)17441 2767 y Fp(k)24 b Fo(=1)19378 -673 y Fm(\022)20489 302 y Fq(i)20933 -655 y Fn(p)p 22039 -655 623 54 v 22039 302 a Fq(r)p 20489 895 2173 54 v 21250 2111 a Fs(2)23090 1200 y Fn(\000)295 b Fs(cos)26381 -673 y Fm(\022)28253 302 y Fq(k)45 b(\031)p 27491 895 3029 54 v 27491 2111 a(n)296 b Fs(+)f(1)30653 -673 y Fm(\023\023)12754 5517 y Fs(=)1107 b Fq(i)15317 4969 y Fp(n)16737 3857 y(n)16173 4255 y Fm(Y)16164 7084 y Fp(k)24 b Fo(=1)18101 3644 y Fm(\022)19079 5517 y Fq(i)19523 4496 y Fn(p)p 20630 4496 623 54 v 1021 x Fq(r)331 b Fn(\000)296 b Fs(2)221 b(cos)25710 3644 y Fm(\022)27582 4619 y Fq(k)45 b(\031)p 26821 5212 3029 54 v 26821 6429 a(n)296 b Fs(+)e(1)29983 3644 y Fm(\023)o(\023)12754 9835 y Fs(=)1107 b Fq(i)15317 9286 y Fp(n)15943 7962 y Fm(\032)17602 8174 y Fp(m)17170 8573 y Fm(Y)17161 11402 y Fp(k)24 b Fo(=1)19098 7962 y Fm(\022)20076 9835 y Fq(i)20520 8813 y Fn(p)p 21626 8813 623 54 v 21626 9835 a Fq(r)332 b Fn(\000)295 b Fs(2)221 b(cos)26707 7962 y Fm(\022)28578 8936 y Fq(k)45 b(\031)p 27817 9529 3029 54 v 27817 10746 a(n)296 b Fs(+)f(1)30979 7962 y Fm(\023\023)221 b(\022)34134 9835 y Fq(i)34578 8813 y Fn(p)p 35684 8813 623 54 v 35684 9835 a Fq(r)332 b Fs(+)295 b(2)221 b(cos)40743 7962 y Fm(\022)42615 8936 y Fq(k)45 b(\031)p 41854 9529 3029 54 v 41854 10746 a(n)295 b Fs(+)g(1)45015 7962 y Fm(\023\023)221 b(\033)48484 9835 y Fn(\001)17474 12174 y Fm(\022)18452 14047 y Fq(i)18896 13025 y Fn(p)p 20003 13025 623 54 v 1022 x Fq(r)331 b Fn(\000)296 b Fs(2)221 b(cos)25083 12174 y Fm(\022)26194 13071 y Fs(\()p Fq(m)294 b Fs(+)29572 12548 y Fo(1)p 29572 12765 471 54 v 29572 13529 a(2)30175 13071 y Fs(\))p Fq(\031)p 26194 13741 5273 54 v 26810 14958 a Fs(2)p Fq(m)h Fs(+)g(1)31599 12174 y Fm(\023\023)12754 17926 y Fs(=)1107 b Fq(i)15317 17378 y Fp(n)16606 16266 y(m)16173 16664 y Fm(Y)16164 19493 y Fp(k)24 b Fo(=1)18101 16053 y Fm(\022)19079 17926 y Fn(\000)p Fq(r)332 b Fn(\000)295 b Fs(4)221 b(cos)24971 17378 y Fo(2)25719 16053 y Fm(\022)27590 17028 y Fq(k)45 b(\031)p 26829 17621 3029 54 v 26829 18837 a(n)296 b Fs(+)f(1)29991 16053 y Fm(\023\023)32168 16452 y(\020)32961 17926 y Fq(i)33405 16905 y Fn(p)p 34512 16905 623 54 v 1021 x Fq(r)331 b Fn(\000)296 b Fs(2)221 b(cos)39592 16452 y Fm(\020)40518 17028 y Fq(\031)p 40518 17621 786 54 v 40586 18837 a Fs(2)41437 16452 y Fm(\021\021)12754 22243 y Fs(=)1107 b Fq(i)15317 21695 y Fo(2)p Fp(m)p Fo(+1)17877 22243 y Fs(\()p Fn(\000)p Fs(1\))20572 21695 y Fp(m)22123 20583 y(m)21690 20982 y Fm(Y)21681 23810 y Fp(k)24 b Fo(=1)23618 20370 y Fm(\022)24596 22243 y Fq(r)332 b Fs(+)294 b(4)221 b(cos)29434 21695 y Fo(2)30181 20370 y Fm(\022)32053 21345 y Fq(k)45 b(\031)p 31291 21938 3029 54 v 31291 23155 a(n)296 b Fs(+)f(1)34453 20370 y Fm(\023\023)36630 20769 y(\020)37423 22243 y Fq(i)37867 21222 y Fn(p)p 38974 21222 623 54 v 1021 x Fq(r)331 b Fn(\000)296 b Fs(2)221 b(cos)44054 20769 y Fm(\020)44981 21345 y Fq(\031)p 44981 21938 786 54 v 45049 23155 a Fs(2)45899 20769 y Fm(\021\021)12754 27073 y Fs(=)14924 25183 y Fl(b)15475 24821 y Fj(n)p 15475 24982 512 40 v 15528 25530 a Fk(2)16119 25183 y Fl(c)14882 25811 y Fm(Y)14873 28640 y Fp(k)24 b Fo(=1)16810 25200 y Fm(\022)17788 27073 y Fq(r)331 b Fs(+)295 b(4)221 b(cos)22625 26524 y Fo(2)23373 25200 y Fm(\022)25244 26174 y Fq(k)45 b(\031)p 24483 26767 3029 54 v 24483 27984 a(n)296 b Fs(+)f(1)27645 25200 y Fm(\023\023)29822 27073 y Fs(\()p Fn(\000)31361 26051 y(p)p 32468 26051 623 54 v 1022 x Fq(r)36 b Fs(\))p Fq(:)800 31134 y Fs(The)434 b(result)f(follo)-36 b(ws)435 b(no)-36 b(w)434 b(from)g(the)f(fact)h(that)12663 34535 y Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)e Fq(i)18532 33986 y Fp(n)19158 34535 y Fq(U)20041 34734 y Fp(n)20889 32662 y Fm(\022)21867 34535 y Fq(i)22444 32679 y Fn(p)p 23551 32679 V 957 x Fq(r)p 22444 34229 1730 54 v 22983 35446 a Fs(2)24306 32662 y Fm(\023)221 b(\032)26501 34535 y Fs(\()p Fn(\000)p Fs(1\))29196 33986 y Fp(n)29956 32679 y Fn(p)p 31063 32679 623 54 v 957 x Fq(r)331 b Fs(+)295 b(1)p 29956 34229 3982 54 v 30757 35452 a(2)31407 34495 y Fn(p)p 32514 34495 623 54 v 957 x Fq(r)34365 34535 y Fs(+)35805 32679 y Fn(p)p 36912 32679 V 957 x Fq(r)331 b Fn(\000)296 b Fs(1)p 35805 34229 4004 54 v 36617 35452 a(2)37267 34495 y Fn(p)p 38374 34495 623 54 v 957 x Fq(r)39941 32662 y Fm(\033)800 38573 y Fs(where)434 b(\()p Fn(\000)p Fs(1\))7253 38091 y Fp(n)8012 37315 y Fl(p)p 8796 37315 451 40 v 639 x Fp(r)26 b Fo(+1)p 8012 38267 2438 54 v 8378 39039 a(2)8848 38400 y Fl(p)p 9632 38400 451 40 v 639 x Fp(r)10877 38573 y Fs(+)12317 37315 y Fl(p)p 13101 37315 V 639 x Fp(r)f Fl(\000)p Fo(1)p 12317 38267 2438 54 v 12682 39039 a(2)13152 38400 y Fl(p)p 13937 38400 451 40 v 13937 39039 a Fp(r)15320 38573 y Fs(is)434 b(the)f(sequence)18480 42440 y(1)p Fq(;)221 b Fn(\000)21418 41541 y Fs(1)p 20878 42134 1730 54 v 20878 42400 a Fn(p)p 21986 42400 623 54 v 21986 43357 a Fq(r)22741 42440 y(;)g Fs(1)p Fq(;)g Fn(\000)26262 41541 y Fs(1)p 25721 42134 1730 54 v 25721 42400 a Fn(p)p 26829 42400 623 54 v 26829 43357 a Fq(r)27585 42440 y(;)g Fs(1)p Fq(;)g Fn(\000)31106 41541 y Fs(1)p 30565 42134 1730 54 v 30565 42400 a Fn(p)p 31673 42400 623 54 v 31673 43357 a Fq(r)32428 42440 y(;)g(:)g(:)g(:)j(:)p 51860 45992 45 878 v 51905 45158 781 45 v 51905 45992 V 52684 45992 45 878 v 2751 48593 a Fs(W)-108 b(e)531 b(note)g(that)g(the)f (case)i Fq(r)571 b Fs(=)534 b(5)e(of)g(this)e(result)h(is)g(noted)g(in) p 0 1 0 0 TeXcolorcmyk 34726 48859 a SDict begin H.S end 34726 48859 a 0 1 0 0 TeXcolorcmyk -266 x Fs(A152119)p 34726 48806 4878 54 v 0 1 0 0 TeXcolorcmyk 39603 47686 a SDict begin H.R end 39603 47686 a 39603 48593 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A152119) >> /Subtype /Link H.B /ANN pdfmark end 39603 48593 a Black 531 w Fs(\(Roger)g(L.)h(Bagula)g(and)800 50199 y(Gary)434 b(W.)g(Adamson\).)800 52532 y SDict begin H.S end 800 52532 a 800 52532 a SDict begin 14.5 H.A end 800 52532 a 800 52532 a SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark end 800 52532 a 2104 x Ft(5)2152 b Fe(b)p Fd(\()p Fe(n)p Fd(;)319 b Fe(r)52 b Fd(\))718 b Ft(and)e(Cheb)-60 b(yshev)715 b(p)60 b(olynomials)800 57556 y Fs(W)-108 b(e)434 b(can)f(also)i(express)e(the)g(second)h (sequence)f(family)i Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))435 b(in)e(terms)h(of)g(Cheb)-36 b(yshev)434 b(p)36 b(olynomials.)800 57888 y SDict begin H.S end 800 57888 a 800 57888 a SDict begin 14.5 H.A end 800 57888 a 800 57888 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.13) cvn H.B /DEST pdfmark end 800 57888 a 800 58995 a SDict begin H.S end 800 58995 a 800 58995 a SDict begin 14.5 H.A end 800 58995 a 800 58995 a SDict begin [ /View [/XYZ H.V] /Dest (equation.18) cvn H.B /DEST pdfmark end 800 58995 a Black 1273 x Fg(Prop)42 b(osition)500 b(13.)p Black 17233 62812 a Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)23203 61151 y Fp(n)22529 61550 y Fm(X)22630 64379 y Fp(k)24 b Fo(=0)24448 62812 y Fs(\()p Fn(\000)p Fs(1\))27143 62263 y Fp(n)p Fl(\000)p Fp(k)29014 62812 y Fq(U)29897 63011 y Fp(k)30688 60939 y Fm(\022)31799 61913 y Fq(r)331 b Fs(+)295 b(2)p 31799 62506 2875 54 v 32911 63723 a(2)34806 60939 y Fm(\023)36005 62812 y Fq(:)14122 b Fs(\(18\))p Black 26150 70521 a(17)p Black eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 1328 x Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(ha)-36 b(v)g(e)6256 -1075 y Fm(\022)11329 -101 y Fs(1)p 7367 492 8576 54 v 7367 1709 a(\(1)295 b Fn(\000)h Fq(x)p Fs(\)\(1)f Fn(\000)g Fq(x)14910 1325 y Fo(2)15436 1709 y Fs(\))16075 798 y Fq(;)18696 -101 y(x)p 16790 492 4551 54 v 16790 1709 a Fs(\(1)h Fn(\000)f Fq(x)p Fs(\))20815 1325 y Fo(2)21474 -1075 y Fm(\023)22747 798 y Fn(\001)25036 -101 y Fs(1)p 23544 492 3636 54 v 23544 1709 a(1)g Fn(\000)g Fq(r)36 b(x)28419 798 y Fs(=)37735 -101 y(1)p 30670 492 14781 54 v 30670 1709 a(\(1)295 b(+)g Fq(x)p Fs(\)\(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)g Fs(+)g Fq(x)44419 1325 y Fo(2)44945 1709 y Fs(\))28419 4257 y(=)38615 3359 y(1)p 30670 3952 16541 54 v 30670 5169 a(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(1\))p Fq(x)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(1\))p Fq(x)43818 4785 y Fo(2)44639 5169 y Fs(+)g Fq(x)46685 4785 y Fo(3)800 7782 y Fs(as)434 b(an)g(easy)h(calculation)g(will)g(v)-36 b(erify)-108 b(.)580 b(This)434 b(implies)g(that)g(for)g(the)f (sequence)h Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))435 b(with)f(generating)800 9387 y(function)15504 11253 y Fq(g)16127 11452 y Fp(b)16584 11253 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)29368 10354 y(1)p 21423 10947 V 21423 12164 a(1)295 b Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)295 b(1\))p Fq(x)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)34570 11780 y Fo(2)35392 12164 y Fs(+)h Fq(x)37438 11780 y Fo(3)800 14183 y Fs(w)-36 b(e)434 b(ha)-36 b(v)g(e)17233 16980 y Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)22682 15090 y Fl(b)23233 14728 y Fj(n)p 23233 14889 512 40 v 23286 15438 a Fk(2)23877 15090 y Fl(c)22529 15718 y Fm(X)22630 18547 y Fp(k)24 b Fo(=0)24448 16980 y Fs(\()p Fn(\000)p Fs(1\))27143 16432 y Fp(n)p Fl(\000)p Fp(k)29014 16980 y Fq(U)29897 17179 y Fp(k)30688 15107 y Fm(\022)31799 16081 y Fq(r)331 b Fs(+)295 b(2)p 31799 16674 2875 54 v 32911 17891 a(2)34806 15107 y Fm(\023)36005 16980 y Fq(:)p 51860 20322 45 878 v 51905 19488 781 45 v 51905 20322 V 52684 20322 45 878 v 800 21274 a SDict begin H.S end 800 21274 a 800 21274 a SDict begin 14.5 H.A end 800 21274 a 800 21274 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.14) cvn H.B /DEST pdfmark end 800 21274 a Black 1628 x Fg(Corollary)521 b(14.)p Black 554 w Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))466 b Fh(is)e(the)h(\014rst)g (di\013er)-66 b(enc)g(e)462 b(of)j(the)f(se)-66 b(quenc)g(e)463 b(with)i(gener)-66 b(al)464 b(term)18313 27206 y Fq(w)36 b Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)24174 25317 y Fl(b)24725 24954 y Fj(n)p 24725 25115 512 40 v 24778 25664 a Fk(2)25369 25317 y Fl(c)24021 25944 y Fm(X)24123 28773 y Fp(k)24 b Fo(=0)26161 27206 y Fq(U)27044 27405 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)29608 25333 y Fm(\022)30719 26308 y Fq(r)331 b Fs(+)295 b(2)p 30719 26901 2875 54 v 31831 28117 a(2)33726 25333 y Fm(\023)34926 27206 y Fq(:)p Black 800 31114 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(ha)-36 b(v)g(e)2282 34832 y Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))1108 b(=)9728 33171 y Fp(n)9054 33570 y Fm(X)9155 36399 y Fp(k)24 b Fo(=0)10973 34832 y Fs(\()p Fn(\000)p Fs(1\))13668 34283 y Fp(n)p Fl(\000)p Fp(k)15539 34832 y Fq(U)16422 35031 y Fp(k)17213 32959 y Fm(\022)18324 33933 y Fq(r)331 b Fs(+)295 b(2)p 18324 34526 V 19436 35743 a(2)21331 32959 y Fm(\023)6935 38900 y Fs(=)1107 b Fq(U)9937 39099 y Fp(n)10785 37027 y Fm(\022)11896 38002 y Fq(r)331 b Fs(+)295 b(2)p 11896 38595 V 13008 39812 a(2)14903 37027 y Fm(\023)16176 38900 y Fn(\000)g Fq(U)18387 39099 y Fp(n)p Fl(\000)p Fo(1)20438 37027 y Fm(\022)21548 38002 y Fq(r)332 b Fs(+)295 b(2)p 21548 38595 V 22660 39812 a(2)24556 37027 y Fm(\023)25829 38900 y Fs(+)g Fq(U)28019 39099 y Fp(n)p Fl(\000)p Fo(2)30069 37027 y Fm(\022)31180 38002 y Fq(r)331 b Fs(+)295 b(2)p 31180 38595 V 32292 39812 a(2)34187 37027 y Fm(\023)35460 38900 y Fn(\000)g Fq(:)221 b(:)g(:)6935 42531 y Fs(=)1107 b Fq(U)9937 42730 y Fp(n)10785 40658 y Fm(\022)11896 41633 y Fq(r)331 b Fs(+)295 b(2)p 11896 42226 V 13008 43442 a(2)14903 40658 y Fm(\023)16176 42531 y Fs(+)g Fq(U)18366 42730 y Fp(n)p Fl(\000)p Fo(2)20416 40658 y Fm(\022)21527 41633 y Fq(r)331 b Fs(+)295 b(2)p 21527 42226 V 22639 43442 a(2)24534 40658 y Fm(\023)25807 42531 y Fs(+)g Fq(:)221 b(:)g(:)297 b Fn(\000)e Fq(U)31147 42730 y Fp(n)p Fl(\000)p Fo(1)33198 40658 y Fm(\022)34308 41633 y Fq(r)332 b Fs(+)294 b(2)p 34308 42226 V 35420 43442 a(2)37316 40658 y Fm(\023)38589 42531 y Fn(\000)h Fq(U)40800 42730 y Fp(n)p Fl(\000)p Fo(3)42850 40658 y Fm(\022)43961 41633 y Fq(r)331 b Fs(+)295 b(2)p 43961 42226 V 45073 43442 a(2)46968 40658 y Fm(\023)48241 42531 y Fn(\000)h Fq(:)221 b(:)g(:)6935 47060 y Fs(=)9207 45170 y Fl(b)9758 44808 y Fj(n)p 9758 44969 512 40 v 9811 45518 a Fk(2)10402 45170 y Fl(c)9054 45798 y Fm(X)9155 48627 y Fp(k)24 b Fo(=0)11194 47060 y Fq(U)12077 47259 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)14641 45187 y Fm(\022)15751 46161 y Fq(r)332 b Fs(+)295 b(2)p 15751 46754 2875 54 v 16864 47971 a(2)18759 45187 y Fm(\023)20032 47060 y Fn(\000)21360 45170 y Fl(b)21911 44797 y Fj(n)p Fi(\000)p Fk(1)p 21911 44969 1557 40 v 22487 45518 a(2)23601 45170 y Fl(c)21730 45798 y Fm(X)21832 48627 y Fp(k)24 b Fo(=0)24240 47060 y Fq(U)25123 47259 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)g Fl(\000)p Fo(1)28889 45187 y Fm(\022)30000 46161 y Fq(r)332 b Fs(+)294 b(2)p 30000 46754 2875 54 v 31112 47971 a(2)33008 45187 y Fm(\023)34207 47060 y Fq(:)p 51860 50923 45 878 v 51905 50089 781 45 v 51905 50923 V 52684 50923 45 878 v 800 51875 a SDict begin H.S end 800 51875 a 800 51875 a SDict begin 14.5 H.A end 800 51875 a 800 51875 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.15) cvn H.B /DEST pdfmark end 800 51875 a Black 1628 x Fg(Corollary)521 b(15.)p Black 9896 57733 a Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)15344 55844 y Fl(b)15895 55482 y Fj(n)p 15895 55643 512 40 v 15948 56191 a Fk(2)16539 55844 y Fl(c)15191 56471 y Fm(X)15293 59300 y Fp(k)24 b Fo(=0)17332 55860 y Fm(\032)18328 57733 y Fq(U)19211 57932 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)g Fo(+1)22977 55860 y Fm(\022)24088 56835 y Fq(r)331 b Fs(+)295 b(2)p 24088 57428 2875 54 v 25200 58644 a(2)27095 55860 y Fm(\023)28368 57733 y Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(U)34465 57932 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)37029 55860 y Fm(\022)38140 56835 y Fq(r)332 b Fs(+)294 b(2)p 38140 57428 V 39252 58644 a(2)41148 55860 y Fm(\023)221 b(\033)43343 57733 y Fq(:)p Black 800 61641 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(use)f(the)g(Cheb)-36 b(yshev)434 b(recurrence)11208 64420 y Fq(U)12091 64619 y Fp(n)p Fo(+1)14289 64420 y Fs(=)368 b(2)p Fq(xU)17941 64619 y Fp(n)18568 64420 y Fs(\()p Fq(x)p Fs(\))295 b Fn(\000)h Fq(U)22826 64619 y Fp(n)p Fl(\000)p Fo(1)24655 64420 y Fs(\()p Fq(x)p Fs(\))p Fq(;)1522 b(U)29172 64619 y Fo(0)29698 64420 y Fs(\()p Fq(x)p Fs(\))369 b(=)f(1)p Fq(;)1523 b(U)36615 64619 y Fo(1)37141 64420 y Fs(\()p Fq(x)p Fs(\))369 b(=)f(2)p Fq(x:)p 51860 67200 45 878 v 51905 66366 781 45 v 51905 67200 V 52684 67200 45 878 v Black 26150 70521 a Fs(18)p Black eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.19) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(W)-108 b(e)434 b(note)f(that)g(the)g (results)g(ab)36 b(o)-36 b(v)g(e)435 b(also)f(follo)-36 b(w)436 b(directly)e(from)g(the)f(observ)-72 b(ation)434 b(that)5439 513 y Fq(g)6062 712 y Fp(b)6519 513 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)18423 -386 y(1)p 11357 208 14781 54 v 11357 1424 a(\(1)296 b(+)e Fq(x)p Fs(\)\(1)i Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)h Fs(+)e Fq(x)25106 1041 y Fo(2)25632 1424 y Fs(\))26640 513 y(=)369 b(\(1)295 b Fn(\000)g Fq(x)p Fs(\))g Fn(\001)34582 -386 y Fs(1)p 33137 208 3539 54 v 33137 1424 a(1)h Fn(\000)f Fq(x)36150 1041 y Fo(2)37104 513 y Fn(\001)42459 -386 y Fs(1)p 37901 208 9767 54 v 37901 1424 a(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)h Fs(+)e Fq(x)47141 1041 y Fo(2)47800 513 y Fq(:)800 3851 y Fs(No)-36 b(w)434 b(the)f(general)h(term)f(of)i(the)e (Riordan)h(arra)-36 b(y)1141 5334 y Fm(\022)6214 6308 y Fs(1)p 2252 6901 8576 54 v 2252 8118 a(\(1)295 b Fn(\000)h Fq(x)p Fs(\)\(1)f Fn(\000)g Fq(x)9795 7734 y Fo(2)10321 8118 y Fs(\))10960 7207 y Fq(;)13581 6308 y(x)p 11676 6901 4551 54 v 11676 8118 a Fs(\(1)g Fn(\000)g Fq(x)p Fs(\))15700 7734 y Fo(2)16359 5334 y Fm(\023)17706 7207 y Fs(=)19086 5334 y Fm(\022)21239 6308 y Fs(1)h(+)f Fq(x)p 20197 6901 5077 54 v 20197 8118 a Fs(\(1)g Fn(\000)h Fq(x)23716 7734 y Fo(2)24241 8118 y Fs(\))24747 7734 y Fo(2)25406 7207 y Fq(;)28027 6308 y(x)p 26121 6901 4551 54 v 26121 8118 a Fs(\(1)g Fn(\000)f Fq(x)p Fs(\))30146 7734 y Fo(2)30805 5334 y Fm(\023)32152 7207 y Fs(=)33532 5334 y Fm(\022)35813 6308 y Fs(1)p 34643 6901 2992 54 v 34643 8118 a(1)g(+)g Fq(x)37767 7207 y(;)221 b(x)39088 5334 y Fm(\023)h(\022)43349 6308 y Fs(1)p 41399 6901 4551 54 v 41399 8118 a(\(1)295 b Fn(\000)g Fq(x)p Fs(\))45423 7734 y Fo(2)46082 7207 y Fq(;)48703 6308 y(x)p 46798 6901 V 46798 8118 a Fs(\(1)g Fn(\000)g Fq(x)p Fs(\))50822 7734 y Fo(2)51481 5334 y Fm(\023)800 10563 y Fs(can)434 b(b)36 b(e)433 b(expressed)g(as)3503 12741 y Fp(n)p Fl(\000)p Fp(k)3452 13139 y Fm(X)3594 15939 y Fp(j)51 b Fo(=0)5370 14401 y Fs(\()p Fn(\000)p Fs(1\))8065 13853 y Fp(j)8552 12528 y Fm(\022)9530 13502 y Fq(n)296 b Fs(+)f Fq(k)340 b Fn(\000)295 b Fq(j)370 b Fs(+)295 b(1)10646 15312 y Fq(n)g Fn(\000)g Fq(k)341 b Fn(\000)295 b Fq(j)17118 12528 y Fm(\023)18465 14401 y Fs(=)19897 12741 y Fp(n)p Fl(\000)p Fp(k)19846 13139 y Fm(X)19988 15939 y Fp(j)51 b Fo(=0)21764 14401 y Fs(\()p Fn(\000)p Fs(1\))24459 13853 y Fp(j)24946 12528 y Fm(\022)25924 13502 y Fq(n)296 b Fs(+)f Fq(k)340 b Fn(\000)295 b Fq(j)370 b Fs(+)295 b(1)27907 15312 y(2)p Fq(k)340 b Fs(+)295 b(1)33512 12528 y Fm(\023)34859 14401 y Fs(=)36914 12741 y Fp(n)36240 13139 y Fm(X)36382 15939 y Fp(j)51 b Fo(=0)38159 14401 y Fs(\()p Fn(\000)p Fs(1\))40854 13853 y Fp(n)p Fl(\000)p Fp(j)42643 12528 y Fm(\022)43621 13502 y Fq(j)370 b Fs(+)295 b Fq(k)340 b Fs(+)295 b(1)44403 15312 y(2)p Fq(k)341 b Fs(+)295 b(1)48809 12528 y Fm(\023)49787 14401 y Fq(:)800 18338 y Fs(This)349 b(is)p 0 1 0 0 TeXcolorcmyk 4907 18603 a SDict begin H.S end 4907 18603 a 0 1 0 0 TeXcolorcmyk -265 x Fs(A158909)p 4907 18550 4878 54 v 0 1 0 0 TeXcolorcmyk 9784 17430 a SDict begin H.R end 9784 17430 a 9784 18338 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A158909) >> /Subtype /Link H.B /ANN pdfmark end 9784 18338 a Black Fs(,)366 b(whic)-36 b(h)348 b(has)g(ro)-36 b(w)348 b(sums)g(giv)-36 b(en)349 b(b)-36 b(y)348 b(the)f(pro)36 b(duct)348 b(of)g(consecutiv) -36 b(e)349 b(Fib)36 b(onacci)349 b(n)-36 b(um)g(b)36 b(ers)800 19943 y Fq(F)181 b Fs(\()p Fq(n)278 b Fs(+)g(1\))p Fq(F)181 b Fs(\()p Fq(n)278 b Fs(+)g(2\))425 b(and)g(diagonal)h(sums)f (giv)-36 b(en)425 b(b)-36 b(y)426 b(the)e(Jacobsthal)i(n)-36 b(um)g(b)36 b(ers)p 0 1 0 0 TeXcolorcmyk 41359 20209 a SDict begin H.S end 41359 20209 a 0 1 0 0 TeXcolorcmyk -266 x Fs(A001045)p 41359 20155 4878 54 v 0 1 0 0 TeXcolorcmyk 46236 19035 a SDict begin H.R end 46236 19035 a 46236 19943 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A001045) >> /Subtype /Link H.B /ANN pdfmark end 46236 19943 a Black Fs(.)575 b(The)425 b(arra)-36 b(y)800 21548 y(b)36 b(egins)17782 21970 y Fm(0)17782 24308 y(B)17782 25105 y(B)17782 25902 y(B)17782 26699 y(B)17782 27496 y(B)17782 28293 y(B)17782 29090 y(B)17782 29887 y(B)17782 30684 y(B)17782 31534 y(@)19498 23123 y Fs(1)1432 b(0)1758 b(0)g(0)1432 b(0)1107 b(0)h Fq(:)221 b(:)g(:)19498 24728 y Fs(1)1432 b(1)1758 b(0)g(0)1432 b(0)1107 b(0)h Fq(:)221 b(:)g(:)19498 26333 y Fs(2)1432 b(3)1758 b(1)g(0)1432 b(0)1107 b(0)h Fq(:)221 b(:)g(:)19498 27938 y Fs(2)1432 b(7)1758 b(5)g(1)1432 b(0)1107 b(0)h Fq(:)221 b(:)g(:)19498 29543 y Fs(3)1107 b(13)h(16)1433 b(7)f(1)1107 b(0)h Fq(:)221 b(:)g(:)19498 31148 y Fs(3)1107 b(22)h(40)g(29)f(9)g(1)h Fq(:)221 b(:)g(:)19643 32423 y Fs(.)19643 32866 y(.)19643 33309 y(.)21725 32423 y(.)21725 32866 y(.)21725 33309 y(.)24133 32423 y(.)24133 32866 y(.)24133 33309 y(.)26540 32423 y(.)26540 32866 y(.)26540 33309 y(.)28623 32423 y(.)28623 32866 y(.)28623 33309 y(.)30380 32423 y(.)30380 32866 y(.)30380 33309 y(.)32067 32534 y(.)32575 32866 y(.)33084 33198 y(.)34073 21970 y Fm(1)34073 24308 y(C)34073 25105 y(C)34073 25902 y(C)34073 26699 y(C)34073 27496 y(C)34073 28293 y(C)34073 29090 y(C)34073 29887 y(C)34073 30684 y(C)34073 31534 y(A)35457 28227 y Fq(:)800 35510 y Fs(W)-108 b(e)434 b(th)-36 b(us)432 b(ha)-36 b(v)g(e)8900 33905 y SDict begin H.S end 8900 33905 a 8900 33905 a SDict begin 14.5 H.A end 8900 33905 a 8900 33905 a SDict begin [ /View [/XYZ H.V] /Dest (equation.19) cvn H.B /DEST pdfmark end 8900 33905 a 5670 39090 a Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)11639 37430 y Fp(n)10965 37828 y Fm(X)11067 40657 y Fp(k)24 b Fo(=0)13157 37430 y Fp(n)p Fl(\000)p Fp(k)13105 37828 y Fm(X)13248 40628 y Fp(j)51 b Fo(=0)15024 39090 y Fs(\()p Fn(\000)p Fs(1\))17719 38542 y Fp(j)18206 37217 y Fm(\022)19184 38192 y Fq(n)295 b Fs(+)g Fq(k)341 b Fn(\000)295 b Fq(j)370 b Fs(+)295 b(1)20299 40002 y Fq(n)h Fn(\000)f Fq(k)340 b Fn(\000)296 b Fq(j)26772 37217 y Fm(\023)27750 39090 y Fq(r)28372 38542 y Fp(k)29310 39090 y Fs(=)31364 37430 y Fp(n)30691 37828 y Fm(X)30792 40657 y Fp(k)24 b Fo(=0)33505 37430 y Fp(n)32831 37828 y Fm(X)32973 40628 y Fp(j)51 b Fo(=0)34749 39090 y Fs(\()p Fn(\000)p Fs(1\))37444 38542 y Fp(n)p Fl(\000)p Fp(j)39234 37217 y Fm(\022)40212 38192 y Fq(j)370 b Fs(+)295 b Fq(k)340 b Fs(+)295 b(1)40994 40002 y(2)p Fq(k)341 b Fs(+)295 b(1)45400 37217 y Fm(\023)46378 39090 y Fq(r)47000 38542 y Fp(k)47569 39090 y Fq(:)2558 b Fs(\(19\))800 43101 y(The)434 b(p)36 b(olynomials)435 b Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(x)p Fs(\))434 b(b)36 b(egin)7070 45767 y(1)p Fq(;)221 b(x)296 b Fs(+)f(1)p Fq(;)221 b(x)12615 45218 y Fo(2)13437 45767 y Fs(+)295 b(3)p Fq(x)h Fs(+)e(2)p Fq(;)221 b(x)19706 45218 y Fo(3)20528 45767 y Fs(+)295 b(5)p Fq(x)23224 45218 y Fo(2)24046 45767 y Fs(+)g(7)p Fq(x)g Fs(+)g(2)p Fq(;)221 b(x)30315 45218 y Fo(4)31137 45767 y Fs(+)295 b(7)p Fq(x)33833 45218 y Fo(3)34654 45767 y Fs(+)g(16)p Fq(x)38000 45218 y Fo(2)38822 45767 y Fs(+)g(13)p Fq(x)h Fs(+)f(3)p Fq(;)221 b(:)g(:)g(:)800 48433 y Fs(W)-108 b(e)434 b(note)f(that)23565 51099 y(1)1107 b(=)f(1)21223 53036 y Fq(x)296 b Fs(+)f(1)1107 b(=)f(1)296 b Fn(\001)f Fs(\(1)g(+)g Fq(x)p Fs(\))17706 54974 y Fq(x)18445 54425 y Fo(2)19266 54974 y Fs(+)g(3)p Fq(x)h Fs(+)f(2)1107 b(=)f(\(1)296 b(+)f Fq(x)p Fs(\)\(2)g(+)g Fq(x)p Fs(\))14189 56911 y Fq(x)14928 56362 y Fo(3)15749 56911 y Fs(+)g(5)p Fq(x)18445 56362 y Fo(2)19266 56911 y Fs(+)g(7)p Fq(x)h Fs(+)f(2)1107 b(=)f(\(2)296 b(+)f Fq(x)p Fs(\)\(1)g(+)g(3)p Fq(x)g Fs(+)g Fq(x)37932 56362 y Fo(2)38458 56911 y Fs(\))9371 58848 y Fq(x)10110 58299 y Fo(4)10931 58848 y Fs(+)g(7)p Fq(x)13627 58299 y Fo(3)14448 58848 y Fs(+)g(16)p Fq(x)17794 58299 y Fo(2)18616 58848 y Fs(+)g(13)p Fq(x)h Fs(+)f(3)1107 b(=)f(\(1)296 b(+)f(3)p Fq(x)g Fs(+)g Fq(x)33929 58299 y Fo(2)34455 58848 y Fs(\)\(3)g(+)g(4)p Fq(x)h Fs(+)e Fq(x)41449 58299 y Fo(2)41975 58848 y Fs(\))221 b Fq(:)g(:)g(:)800 61514 y Fs(Alternativ)-36 b(e)449 b(\(and)f(more)g(esoteric\))h (expressions)f(can)h(b)36 b(e)448 b(found)g(for)h Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))449 b(b)-36 b(y)449 b(taking)g(other)f(factor-)800 63119 y(izations)434 b(of)h(the)e(co)36 b(e\016cien)-36 b(t)433 b(arra)-36 b(y)-108 b(.)579 b(F)-108 b(or)433 b(instance,)5991 64729 y Fm(\022)11064 65704 y Fs(1)p 7101 66297 8576 54 v 7101 67513 a(\(1)296 b Fn(\000)f Fq(x)p Fs(\)\(1)g Fn(\000)h Fq(x)14645 67130 y Fo(2)15171 67513 y Fs(\))15809 66602 y Fq(;)18430 65704 y(x)p 16525 66297 4551 54 v 16525 67513 a Fs(\(1)f Fn(\000)g Fq(x)p Fs(\))20549 67130 y Fo(2)21208 64729 y Fm(\023)22555 66602 y Fs(=)23935 64729 y Fm(\022)26996 65704 y Fs(1)p 25046 66297 V 25046 67513 a(\(1)g Fn(\000)h Fq(x)p Fs(\))29071 67130 y Fo(2)29729 66602 y Fq(;)32350 65704 y(x)p 30445 66297 V 30445 67513 a Fs(\(1)f Fn(\000)g Fq(x)p Fs(\))34469 67130 y Fo(2)35128 64729 y Fm(\023)36401 66602 y Fn(\001)37065 64729 y Fm(\022)38176 65704 y Fs(1)g(+)40428 64655 y Fn(p)p 41535 64655 3642 54 v 1049 x Fs(1)h(+)f(4)p Fq(x)p 38176 66297 7002 54 v 38977 67610 a Fs(2)39627 66562 y Fn(p)p 40734 66562 3642 54 v 1048 x Fs(1)h(+)f(4)p Fq(x)45310 66602 y(;)221 b(x)46631 64729 y Fm(\023)p Black 26150 70521 a Fs(19)p Black eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.20) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(leads)434 b(to)g(the)f(expression)11179 504 y Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)17148 -1156 y Fp(n)16474 -757 y Fm(X)16576 2071 y Fp(k)24 b Fo(=0)19288 -1156 y Fp(n)18615 -757 y Fm(X)18757 2042 y Fp(j)51 b Fo(=0)20755 -1368 y Fm(\022)21732 -394 y Fq(n)296 b Fs(+)f Fq(j)370 b Fs(+)295 b(1)22848 1416 y Fq(n)g Fn(\000)h Fq(j)26976 -1368 y Fm(\023\022)28932 -394 y Fs(2)p Fq(j)370 b Fn(\000)295 b Fs(2)p Fq(k)341 b Fn(\000)295 b Fs(1)30719 1416 y Fq(j)370 b Fn(\000)295 b Fq(k)35464 -1368 y Fm(\023)36441 504 y Fs(\()p Fn(\000)p Fs(1\))39136 -44 y Fp(j)51 b Fl(\000)p Fp(k)40869 504 y Fq(r)41491 -44 y Fp(k)42060 504 y Fq(:)800 4246 y SDict begin H.S end 800 4246 a 800 4246 a SDict begin 14.5 H.A end 800 4246 a 800 4246 a SDict begin [ /View [/XYZ H.V] /Dest (section.6) cvn H.B /DEST pdfmark end 800 4246 a 1544 x Ft(6)2152 b(F)-179 b(rom)717 b Fe(b)p Fd(\()p Fe(n)p Fd(;)319 b Fe(r)52 b Fd(\))718 b Ft(to)f Fe(a)p Fd(\()p Fe(n)p Fd(;)319 b Fe(r)52 b Fd(\))800 8710 y Fs(W)-108 b(e)440 b(no)-36 b(w)439 b(sho)-36 b(w)440 b(that)f(w)-36 b(e)440 b(can)g(\014nd)e(a)i(transformation)g(from)g(the)f(sequences)h Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))440 b(with)g(generating)800 10315 y(function)15504 12181 y Fq(g)16127 12380 y Fp(b)16584 12181 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)29368 11282 y(1)p 21423 11875 16541 54 v 21423 13092 a(1)295 b Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)295 b(1\))p Fq(x)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)34570 12708 y Fo(2)35392 13092 y Fs(+)h Fq(x)37438 12708 y Fo(3)800 15219 y Fs(to)434 b(the)f(sequences)g Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))435 b(with)e(generating)h(function)f Fq(g)29681 15418 y Fp(a)30236 15219 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)36497 14696 y Fo(1+)p Fp(x)p Fl(\000)p Fp(x)39491 14384 y Fk(2)p 35074 14914 6301 54 v 35074 15678 a Fo(1)p Fl(\000)p Fo(\()p Fp(r)26 b Fo(+2\))p Fp(x)39191 15425 y Fk(2)39652 15678 y Fo(+)p Fp(x)40914 15425 y Fk(4)41508 15219 y Fs(.)578 b(This)434 b(follo)-36 b(ws)436 b(since)800 18594 y Fq(g)1423 18793 y Fp(a)1978 18594 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))1108 b(=)10498 17695 y(1)296 b(+)e Fq(x)i Fn(\000)f Fq(x)15852 17213 y Fo(2)p 8292 18288 10292 54 v 8292 19505 a Fs(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)15191 19121 y Fo(2)16012 19505 y Fs(+)g Fq(x)18058 19121 y Fo(4)6041 22231 y Fs(=)8159 20358 y Fm(\022)9270 21332 y Fs(1)g(+)g Fq(x)h Fn(\000)f Fq(x)14624 20850 y Fo(2)p 9270 21926 5880 54 v 9672 23142 a Fs(\(1)g Fn(\000)g Fq(x)13190 22759 y Fo(2)13716 23142 y Fs(\))14222 22759 y Fo(2)15283 22231 y Fq(;)17904 21332 y(x)18643 20850 y Fo(2)p 15998 21926 5077 54 v 15998 23142 a Fs(\(1)h Fn(\000)f Fq(x)19517 22759 y Fo(2)20043 23142 y Fs(\))20549 22759 y Fo(2)21207 20358 y Fm(\023)22480 22231 y Fn(\001)24770 21332 y Fs(1)p 23277 21926 3636 54 v 23277 23142 a(1)h Fn(\000)f Fq(r)36 b(x)6041 26082 y Fs(=)8159 24209 y Fm(\022)9270 25183 y Fs(1)295 b(+)g Fq(x)h Fn(\000)f Fq(x)14624 24701 y Fo(2)p 9270 25777 5880 54 v 9672 26993 a Fs(\(1)g Fn(\000)g Fq(x)13190 26610 y Fo(2)13716 26993 y Fs(\))14222 26610 y Fo(2)15283 26082 y Fq(;)17904 25183 y(x)18643 24701 y Fo(2)p 15998 25777 5077 54 v 15998 26993 a Fs(\(1)h Fn(\000)f Fq(x)19517 26610 y Fo(2)20043 26993 y Fs(\))20549 26610 y Fo(2)21207 24209 y Fm(\023)22480 26082 y Fn(\001)23144 24209 y Fm(\022)28217 25183 y Fs(1)p 24255 25777 8576 54 v 24255 26993 a(\(1)g Fn(\000)h Fq(x)p Fs(\)\(1)f Fn(\000)g Fq(x)31798 26610 y Fo(2)32324 26993 y Fs(\))32963 26082 y Fq(;)35584 25183 y(x)p 33678 25777 4551 54 v 33678 26993 a Fs(\(1)h Fn(\000)f Fq(x)p Fs(\))37703 26610 y Fo(2)38362 24209 y Fm(\023)39340 24507 y Fl(\000)p Fo(1)40892 26082 y Fn(\001)49635 25183 y Fs(1)p 41689 25777 16541 54 v 41689 26993 a(1)h Fn(\000)f Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)i Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(1\))p Fq(x)54837 26610 y Fo(2)55659 26993 y Fs(+)f Fq(x)57704 26610 y Fo(3)6041 29738 y Fs(=)8159 27865 y Fm(\022)9270 28840 y Fs(1)h(+)g Fq(x)h Fn(\000)f Fq(x)14624 28358 y Fo(2)p 9270 29433 5880 54 v 9672 30649 a Fs(\(1)g Fn(\000)g Fq(x)13190 30266 y Fo(2)13716 30649 y Fs(\))14222 30266 y Fo(2)15283 29738 y Fq(;)17904 28840 y(x)18643 28358 y Fo(2)p 15998 29433 5077 54 v 15998 30649 a Fs(\(1)h Fn(\000)f Fq(x)19517 30266 y Fo(2)20043 30649 y Fs(\))20549 30266 y Fo(2)21207 27865 y Fm(\023)22480 29738 y Fn(\001)g Fs(\()p Fq(c)p Fs(\()p Fn(\000)p Fq(x)p Fs(\))26994 29190 y Fo(2)27815 29738 y Fs(+)g Fq(xc)p Fs(\()p Fn(\000)p Fq(x)p Fs(\))33205 29190 y Fo(4)33730 29738 y Fq(;)221 b Fs(1)297 b Fn(\000)e Fq(c)p Fs(\()p Fn(\000)p Fq(x)p Fs(\)\))g Fn(\001)49474 28840 y Fs(1)p 41529 29433 16541 54 v 41529 30649 a(1)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)295 b(1\))p Fq(x)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)54676 30266 y Fo(2)55498 30649 y Fs(+)h Fq(x)57544 30266 y Fo(3)6041 33198 y Fs(=)1106 b(\(1)296 b(+)e Fq(x)i Fs(+)f Fq(x)13998 32649 y Fo(3)14819 33198 y Fn(\000)g Fq(x)16886 32649 y Fo(4)17412 33198 y Fq(;)221 b(x)18733 32649 y Fo(2)19260 33198 y Fs(\))295 b Fn(\001)28803 32299 y Fs(1)p 20858 32892 V 20858 34109 a(1)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)i Fn(\000)f Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)34005 33725 y Fo(2)34827 34109 y Fs(+)g Fq(x)36872 33725 y Fo(3)6041 36003 y Fs(=)1106 b(\(1)296 b(+)e Fq(x)i Fs(+)f Fq(x)13998 35454 y Fo(3)14819 36003 y Fn(\000)g Fq(x)16886 35454 y Fo(4)17412 36003 y Fq(;)221 b(x)18733 35454 y Fo(2)19260 36003 y Fs(\))295 b Fn(\001)g Fq(g)21348 36202 y Fp(b)21805 36003 y Fs(\()p Fq(x)p Fs(;)221 b Fq(r)36 b Fs(\))p Fq(:)800 38241 y Fs(Here,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(used)e(the)h(notation)21115 41291 y Fq(c)p Fs(\()p Fq(x)p Fs(\))369 b(=)25308 40392 y(1)296 b Fn(\000)27582 39348 y(p)p 28689 39348 3664 54 v 1044 x Fs(1)f Fn(\000)h Fs(4)p Fq(x)p 25308 40985 7045 54 v 28135 42202 a Fs(2)p Fq(x)800 43867 y Fs(for)434 b(the)f(generating)h (function)f(of)i(the)e(Catalan)h(n)-36 b(um)g(b)36 b(ers)p 0 1 0 0 TeXcolorcmyk 30361 44133 a SDict begin H.S end 30361 44133 a 0 1 0 0 TeXcolorcmyk -266 x Fs(A000108)p 30361 44080 4878 54 v 0 1 0 0 TeXcolorcmyk 35238 42960 a SDict begin H.R end 35238 42960 a 35238 43867 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000108) >> /Subtype /Link H.B /ANN pdfmark end 35238 43867 a Black Fs(.)800 47078 y(F)-108 b(or)433 b(instance,)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)12164 48132 y Fm(0)12164 50470 y(B)12164 51267 y(B)12164 52064 y(B)12164 52861 y(B)12164 53658 y(B)12164 54455 y(B)12164 55252 y(B)12164 56049 y(B)12164 56846 y(B)12164 57697 y(@)14397 49285 y Fs(1)1624 b(0)1107 b(0)g(0)h(0)f(0)g Fq(:)221 b(:)g(:)14397 50890 y Fs(1)1624 b(0)1107 b(0)g(0)h(0)f(0)g Fq(:)221 b(:)g(:)14397 52495 y Fs(0)1624 b(1)1107 b(0)g(0)h(0)f(0)g Fq(:)221 b(:)g(:)14397 54100 y Fs(1)1624 b(1)1107 b(0)g(0)h(0)f(0)g Fq(:)221 b(:)g(:)13880 55705 y Fn(\000)p Fs(1)1108 b(0)f(1)g(0)h(0)f(0) g Fq(:)221 b(:)g(:)14397 57310 y Fs(0)1624 b(1)1107 b(1)g(0)h(0)f(0)g Fq(:)221 b(:)g(:)14541 58585 y Fs(.)14541 59028 y(.)14541 59471 y(.)16815 58585 y(.)16815 59028 y(.)16815 59471 y(.)18572 58585 y(.)18572 59028 y(.)18572 59471 y(.)20330 58585 y(.)20330 59028 y(.)20330 59471 y(.)22087 58585 y(.)22087 59028 y(.)22087 59471 y(.)23844 58585 y(.)23844 59028 y(.)23844 59471 y(.)25531 58696 y(.)26040 59028 y(.)26549 59360 y(.)27537 48132 y Fm(1)27537 50470 y(C)27537 51267 y(C)27537 52064 y(C)27537 52861 y(C)27537 53658 y(C)27537 54455 y(C)27537 55252 y(C)27537 56049 y(C)27537 56846 y(C)27537 57697 y(A)28921 48132 y(0)28921 50470 y(B)28921 51267 y(B)28921 52064 y(B)28921 52861 y(B)28921 53658 y(B)28921 54455 y(B)28921 55252 y(B)28921 56049 y(B)28921 56846 y(B)28921 57697 y(@)31612 49285 y Fs(1)31612 50890 y(5)31287 52495 y(30)30962 54100 y(174)30637 55705 y(1015)30637 57310 y(5915)31757 58585 y(.)31757 59028 y(.)31757 59471 y(.)33792 48132 y Fm(1)33792 50470 y(C)33792 51267 y(C)33792 52064 y(C)33792 52861 y(C)33792 53658 y(C)33792 54455 y(C)33792 55252 y(C)33792 56049 y(C)33792 56846 y(C)33792 57697 y(A)35323 54389 y Fs(=)36703 48132 y Fm(0)36703 50470 y(B)36703 51267 y(B)36703 52064 y(B)36703 52861 y(B)36703 53658 y(B)36703 54455 y(B)36703 55252 y(B)36703 56049 y(B)36703 56846 y(B)36703 57697 y(@)38744 49285 y Fs(1)38744 50890 y(1)38744 52495 y(5)38744 54100 y(6)38419 55705 y(29)38419 57310 y(35)38889 58585 y(.)38889 59028 y(.)38889 59471 y(.)40273 48132 y Fm(1)40273 50470 y(C)40273 51267 y(C)40273 52064 y(C)40273 52861 y(C)40273 53658 y(C)40273 54455 y(C)40273 55252 y(C)40273 56049 y(C)40273 56846 y(C)40273 57697 y(A)800 62041 y Fs(whic)-36 b(h)433 b(displa)-36 b(ys)434 b(the)g(sequence)f(with)h(g.f.)22990 61518 y Fo(1+)p Fp(x)p Fl(\000)p Fp(x)25984 61206 y Fk(2)p 22524 61736 4387 54 v 22524 62499 a Fo(1)p Fl(\000)p Fo(6)p Fp(x)24726 62247 y Fk(2)25188 62499 y Fo(+)p Fp(x)26450 62247 y Fk(4)27477 62041 y Fs(and)f(general)h(term)18648 66022 y Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b(4\))370 b(=)24254 64132 y Fl(b)24805 63770 y Fj(n)p 24805 63931 512 40 v 24858 64479 a Fk(2)25449 64132 y Fl(c)24101 64760 y Fm(X)24244 67559 y Fp(j)51 b Fo(=0)26241 64149 y Fm(\022)27219 65123 y Fq(n)296 b Fn(\000)f Fq(j)28419 66933 y(j)30232 64149 y Fm(\023)31210 66022 y Fs(4)31860 65473 y Fl(b)32411 65063 y Fj(n)p Fi(\000)p Fk(2)p Fj(j)p 32411 65272 1935 40 v 33176 65820 a Fk(2)34479 65473 y Fl(c)p Black 26150 70521 a Fs(20)p Black eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(as)434 b(the)f(image)h(of)h(the)e (sequence)g(with)h(g.f.)25774 -3195 y Fo(1)p 22950 -2977 6119 54 v 22950 -2214 a(1)p Fl(\000)p Fo(5)p Fp(x)p Fl(\000)p Fo(5)p Fp(x)26884 -2466 y Fk(2)27346 -2214 y Fo(+)p Fp(x)28608 -2466 y Fk(3)29635 -2672 y Fs(and)f(general)h(term)19128 -4 y Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b(4\))370 b(=)e Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b(4\))p Fq(a)p Fs(\()p Fq(n)296 b Fs(+)f(1;)221 b(4\))p Fq(:)800 2665 y Fs(W)-108 b(e)434 b(th)-36 b(us)432 b(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)436 b(prop)36 b(osition.)800 2923 y SDict begin H.S end 800 2923 a 800 2923 a SDict begin 14.5 H.A end 800 2923 a 800 2923 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.16) cvn H.B /DEST pdfmark end 800 2923 a Black 2226 x Fg(Prop)42 b(osition)500 b(16.)p Black 554 w Fh(The)465 b(se)-66 b(quenc)g(e)463 b Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))466 b Fh(with)e(gener)-66 b(ating)463 b(function)26475 7511 y Fs(1)p 18530 8104 16541 54 v 18530 9321 a(1)295 b Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)295 b(1\))p Fq(x)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)31677 8937 y Fo(2)32499 9321 y Fs(+)h Fq(x)34545 8937 y Fo(3)800 12090 y Fh(is)590 b(tr)-66 b(ansforme)g(d)588 b(into)h(the)h(se)-66 b(quenc)g(e)588 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))591 b Fh(with)f(gener)-66 b(ating)587 b(function)40549 11567 y Fo(1+)p Fp(x)p Fl(\000)p Fp(x)43543 11255 y Fk(2)p 39126 11785 6301 54 v 39126 12548 a Fo(1)p Fl(\000)p Fo(\()p Fp(r)26 b Fo(+2\))p Fp(x)43243 12296 y Fk(2)43704 12548 y Fo(+)p Fp(x)44966 12296 y Fk(4)46149 12090 y Fh(and)590 b(gener)-66 b(al)800 13695 y(term)18676 16493 y Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)24254 14603 y Fl(b)24805 14241 y Fj(n)p 24805 14402 512 40 v 24858 14950 a Fk(2)25449 14603 y Fl(c)24101 15231 y Fm(X)24244 18030 y Fp(j)51 b Fo(=0)26241 14620 y Fm(\022)27219 15594 y Fq(n)296 b Fn(\000)f Fq(j)28419 17404 y(j)30232 14620 y Fm(\023)31210 16493 y Fq(r)31832 15944 y Fl(b)32383 15534 y Fj(n)p Fi(\000)p Fk(2)p Fj(j)p 32383 15743 1935 40 v 33148 16291 a Fk(2)34451 15944 y Fl(c)800 20048 y Fh(by)465 b(the)f(\(gener)-66 b(alize)g(d\))464 b(R)-33 b(ior)-66 b(dan)464 b(arr)-66 b(ay)20816 22716 y Fs(\(1)296 b(+)e Fq(x)i Fs(+)f Fq(x)26655 22167 y Fo(3)27476 22716 y Fn(\000)g Fq(x)29543 22167 y Fo(4)30069 22716 y Fq(;)221 b(x)31390 22167 y Fo(2)31917 22716 y Fs(\))p Fq(:)2751 25384 y Fs(W)-108 b(e)436 b(note)g(that)g (this)g(transformation)g(is)h(not)f(in)-36 b(v)g(ertible.)586 b(In)436 b(terms)f(of)i(in)-36 b(v)g(ertible)437 b(transforms,)g(w)-36 b(e)800 26989 y(ha)g(v)g(e)434 b(the)f(follo)-36 b(wing)436 b(result.)800 27247 y SDict begin H.S end 800 27247 a 800 27247 a SDict begin 14.5 H.A end 800 27247 a 800 27247 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.17) cvn H.B /DEST pdfmark end 800 27247 a Black 2227 x Fg(Prop)42 b(osition)570 b(17.)p Black 586 w Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))522 b Fh(is)f(the)h(image)e(under)h(the)h(R)-33 b(ior)-66 b(dan)520 b(arr)-66 b(ay)521 b Fs(\(1)337 b(+)g Fq(x)g Fs(+)g Fq(x)43675 28992 y Fo(3)44538 29474 y Fn(\000)g Fq(x)46647 28992 y Fo(4)47173 29474 y Fq(;)221 b(x)p Fs(\))522 b Fh(of)f(the)800 31079 y(aer)-66 b(ation)464 b(of)g Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))p Fh(.)p Black 800 33563 a(Pr)-66 b(o)g(of.)p Black 649 w Fs(W)-108 b(e)434 b(ha)-36 b(v)g(e)7381 36048 y(1)295 b(+)g Fq(x)g Fn(\000)h Fq(x)12735 35566 y Fo(2)p 5175 36641 10292 54 v 5175 37858 a Fs(1)f Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)12074 37474 y Fo(2)12895 37858 y Fs(+)g Fq(x)14941 37474 y Fo(4)16706 36947 y Fs(=)22501 36048 y(1)g(+)g Fq(x)g Fs(+)g Fq(x)27833 35566 y Fo(3)28654 36048 y Fn(\000)h Fq(x)30722 35566 y Fo(4)p 18958 36641 15833 54 v 18958 37858 a Fs(\(1)f(+)g Fq(x)22455 37474 y Fo(2)22981 37858 y Fs(\)\(1)g Fn(\000)g Fs(\()p Fq(r)332 b Fs(+)294 b(2\))p Fq(x)30891 37474 y Fo(2)31713 37858 y Fs(+)h Fq(x)33759 37474 y Fo(4)34285 37858 y Fs(\))16706 40387 y(=)1107 b(\(1)295 b(+)g Fq(x)g Fs(+)g Fq(x)24663 39839 y Fo(3)25484 40387 y Fn(\000)h Fq(x)27552 39839 y Fo(4)28078 40387 y Fq(;)221 b(x)p Fs(\))295 b Fn(\001)39206 39489 y Fs(1)p 30997 40082 17067 54 v 30997 41299 a(1)h Fn(\000)f Fs(\()p Fq(r)332 b Fs(+)294 b(1\))p Fq(x)37896 40915 y Fo(2)38718 41299 y Fn(\000)h Fs(\()p Fq(r)331 b Fs(+)295 b(1\))p Fq(x)44671 40915 y Fo(4)45492 41299 y Fs(+)g Fq(x)47538 40915 y Fo(6)48197 40387 y Fq(:)p 51860 43682 45 878 v 51905 42849 781 45 v 51905 43682 V 52684 43682 45 878 v 800 44603 a SDict begin H.S end 800 44603 a 800 44603 a SDict begin 14.5 H.A end 800 44603 a 800 44603 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.18) cvn H.B /DEST pdfmark end 800 44603 a Black 1564 x Fg(Corollary)769 b(18.)p Black 665 w Fh(The)663 b(aer)-66 b(ation)663 b(of)g Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))665 b Fh(is)f(the)f(image)g(of)g Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))665 b Fh(under)e(the)h(R)-33 b(ior)-66 b(dan)662 b(arr)-66 b(ay)800 46696 y Fm(\000)3895 47249 y Fo(1)p 1542 47466 5178 54 v 1542 48230 a(1+)p Fp(x)p Fo(+)p Fp(x)4536 47978 y Fk(3)4996 48230 y Fl(\000)p Fp(x)6258 47978 y Fk(4)6852 47772 y Fq(;)221 b(x)8173 46696 y Fm(\001)8783 47772 y Fh(.)2751 50442 y Fs(W)-108 b(e)349 b(no)-36 b(w)349 b(note)g(that)g(the)f(Riordan)h(arra)-36 b(y)23457 49366 y Fm(\000)26552 49919 y Fo(1)p 24199 50136 V 24199 50900 a(1+)p Fp(x)p Fo(+)p Fp(x)27193 50648 y Fk(3)27653 50900 y Fl(\000)p Fp(x)28915 50648 y Fk(4)29509 50442 y Fq(;)221 b(x)30830 49366 y Fm(\001)31789 50442 y Fs(is)349 b(the)f(sequence)h(arra)-36 b(y)350 b(for)g(the)e(sequence) 800 52047 y(with)434 b(general)g(term)19596 53213 y Fl(b)20147 52851 y Fj(n)p 20147 53012 512 40 v 20200 53560 a Fk(2)20791 53213 y Fl(c)19443 53840 y Fm(X)19585 56640 y Fp(j)51 b Fo(=0)21362 55102 y Fs(\()p Fn(\000)p Fs(1\))24057 54554 y Fp(n)p Fl(\000)p Fp(j)25846 55102 y Fq(F)181 b Fs(\()p Fq(n)295 b Fn(\000)h Fs(2)p Fq(j)370 b Fs(+)295 b(1\))p Fq(:)800 58584 y Fs(Hence)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)800 58584 y SDict begin H.S end 800 58584 a 800 58584 a SDict begin 14.5 H.A end 800 58584 a 800 58584 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.19) cvn H.B /DEST pdfmark end 800 58584 a 800 59463 a SDict begin H.S end 800 59463 a 800 59463 a SDict begin 14.5 H.A end 800 59463 a 800 59463 a SDict begin [ /View [/XYZ H.V] /Dest (equation.20) cvn H.B /DEST pdfmark end 800 59463 a Black 1605 x Fg(Corollary)521 b(19.)p Black 10963 65357 a Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))370 b(=)16697 63697 y Fo(2)p Fp(n)16259 64096 y Fm(X)16360 66924 y Fp(k)24 b Fo(=0)18399 63468 y Fl(b)18950 63094 y Fk(2)p Fj(n)p Fi(\000)p Fj(k)p 18950 63266 2010 40 v 19752 63815 a Fk(2)21092 63468 y Fl(c)18995 64096 y Fm(X)19138 66895 y Fp(j)51 b Fo(=0)21510 65357 y Fs(\()p Fn(\000)p Fs(1\))24205 64809 y Fo(2)p Fp(n)p Fl(\000)p Fp(k)24 b Fl(\000)p Fp(j)27711 65357 y Fq(F)181 b Fs(\(2)p Fq(n)295 b Fn(\000)g Fq(k)341 b Fn(\000)295 b Fs(2)p Fq(j)370 b Fs(+)295 b(1\))p Fq(a)p Fs(\()p Fq(k)45 b Fs(;)221 b Fq(r)36 b Fs(\))p Fq(:)7852 b Fs(\(20\))p Black 26150 70521 a(21)p Black eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.22) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (section.7) cvn H.B /DEST pdfmark end 800 -4000 a 1328 x Ft(7)2152 b(A)717 b(Cheb)-60 b(yshev)716 b(pro)60 b(duct)715 b(result)800 249 y Fs(In)433 b(this)h(section,)g(w)-36 b(e)434 b(observ)-36 b(e)433 b(that)h(the)f(p)36 b(olynomials)18605 4519 y Fq(w)g Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))369 b(=)24465 2630 y Fl(b)25016 2268 y Fj(n)p 25016 2429 512 40 v 25069 2977 a Fk(2)25660 2630 y Fl(c)24313 3257 y Fm(X)24414 6086 y Fp(k)24 b Fo(=0)26453 4519 y Fq(U)27336 4718 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)29900 2646 y Fm(\022)31010 3621 y Fq(r)332 b Fs(+)295 b(2)p 31010 4214 2875 54 v 32122 5431 a(2)34018 2646 y Fm(\023)800 8642 y Fs(exhibit)335 b(a)h(m)-36 b(ultiplicativ)g(e)336 b(prop)36 b(ert)-36 b(y)-108 b(.)545 b(T)-108 b(o)335 b(this)g(end,)355 b(w)-36 b(e)335 b(let)h Fq(U)31451 8160 y Fp(d)31312 8970 y(n)31990 8642 y Fs(\()p Fq(x)p Fs(\))f(represen)-36 b(t)334 b(a)i(\\doubled")f (Cheb)-36 b(yshev)800 10247 y(p)36 b(olynomial,)435 b(de\014ned)e(b)-36 b(y)13448 13725 y Fq(U)14470 13177 y Fp(d)14331 14054 y(n)15010 13725 y Fs(\()p Fq(x)p Fs(\))369 b(=)f Fq(U)19526 13562 y Fj(n)p 19526 13723 512 40 v 19579 14272 a Fk(2)20226 13725 y Fs(\()p Fq(x)p Fs(\))22110 12827 y(1)295 b(+)g(\()p Fn(\000)p Fs(1\))27057 12345 y Fp(n)p 22110 13420 5574 54 v 24571 14637 a Fs(2)28111 13725 y(+)g Fq(U)30434 13695 y Fj(n)p Fi(\000)p Fk(1)p 30434 13867 1557 40 v 31009 14417 a(2)32179 13725 y Fs(\()p Fq(x)p Fs(\))34063 12827 y(1)g Fn(\000)g Fs(\()p Fn(\000)p Fs(1\))39031 12345 y Fp(n)p 34063 13420 5596 54 v 36535 14637 a Fs(2)39791 13725 y Fq(:)800 16885 y Fs(Clearly)-108 b(,)15335 19940 y Fq(U)16357 19392 y Fp(d)16218 20269 y(n)16896 19940 y Fs(\()p Fq(x)p Fs(\))369 b(=)20550 18051 y Fl(b)21101 17688 y Fj(n)p 21101 17849 512 40 v 21154 18398 a Fk(4)21745 18051 y Fl(c)20397 18678 y Fm(X)20498 21507 y Fp(k)24 b Fo(=0)22537 18067 y Fm(\022)23515 19042 y Fn(b)24238 18519 y Fp(n)p 24238 18736 571 54 v 24288 19500 a Fo(2)24942 19042 y Fn(c)295 b(\000)h Fq(k)25335 20851 y(k)27877 18067 y Fm(\023)28854 19940 y Fs(\()p Fn(\000)p Fs(1\))31549 19392 y Fp(k)32118 19940 y Fs(\(2)p Fq(x)p Fs(\))34519 19392 y Fl(b)35070 19030 y Fj(n)p 35071 19191 512 40 v 35124 19739 a Fk(2)35715 19392 y Fl(c\000)p Fo(2)p Fp(k)37904 19940 y Fq(:)800 23519 y Fs(The)434 b(generating)f(function) h(for)g(the)f(p)36 b(olynomials)435 b Fq(U)27409 23036 y Fp(d)27270 23847 y(n)27949 23519 y Fs(\()p Fq(x)p Fs(\))e(is)h(then)e (giv)-36 b(en)435 b(b)-36 b(y)25152 26032 y(1)296 b(+)f Fq(y)p 22779 26625 7681 54 v 22779 27842 a Fs(1)h Fn(\000)f Fs(2)p Fq(xy)27124 27458 y Fo(2)27945 27842 y Fs(+)g Fq(y)29934 27458 y Fo(4)30593 26931 y Fq(:)800 30895 y Fs(The)434 b(generating)f(function)h(of)16408 29898 y Fm(P)17810 30090 y Fl(b)18361 29728 y Fj(n)p 18361 29889 512 40 v 18414 30437 a Fk(2)19005 30090 y Fl(c)17810 31265 y Fp(k)24 b Fo(=0)19803 30895 y Fq(U)20686 31094 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)23028 30895 y Fs(\()p Fq(x)p Fs(\))433 b(is)h(similarly)i(giv)-36 b(en)434 b(b)-36 b(y)19599 33446 y(1)p 13935 34039 11978 54 v 13935 35256 a(\(1)295 b Fn(\000)h Fq(y)17397 34872 y Fo(2)17922 35256 y Fs(\)\(1)g Fn(\000)f Fs(2)p Fq(x)h Fs(+)e Fq(y)24881 34872 y Fo(2)25407 35256 y Fs(\))26415 34345 y(=)33291 33446 y(1)p 27928 34039 11376 54 v 27928 35256 a(1)i Fn(\000)f Fs(2)p Fq(xy)344 b Fs(+)294 b(2)p Fq(xy)35946 34872 y Fo(3)36768 35256 y Fn(\000)h Fq(y)38778 34872 y Fo(4)39436 34345 y Fq(:)800 36803 y SDict begin H.S end 800 36803 a 800 36803 a SDict begin 14.5 H.A end 800 36803 a 800 36803 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.20) cvn H.B /DEST pdfmark end 800 36803 a Black 1033 x Fg(Theorem)499 b(20.)p Black 554 w Fh(We)465 b(have)14946 36231 y SDict begin H.S end 14946 36231 a 14946 36231 a SDict begin 14.5 H.A end 14946 36231 a 14946 36231 a SDict begin [ /View [/XYZ H.V] /Dest (equation.21) cvn H.B /DEST pdfmark end 14946 36231 a 18452 38744 a Fl(b)19003 38382 y Fj(n)p 19003 38543 512 40 v 19056 39091 a Fk(2)19647 38744 y Fl(c)18299 39372 y Fm(X)18400 42200 y Fp(k)24 b Fo(=0)20439 40633 y Fq(U)21322 40832 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)23664 40633 y Fs(\()p Fq(x)p Fs(\))369 b(=)g Fq(U)28187 40085 y Fp(d)28048 40962 y(n)28726 40633 y Fs(\()p Fq(x)p Fs(\))p Fq(U)31499 40085 y Fp(d)31360 40962 y(n)p Fo(+1)33189 40633 y Fs(\()p Fq(x)p Fs(\))p Fq(:)15187 b Fs(\(21\))p Black 800 44376 a Fh(Pr)-66 b(o)g(of.)p Black 649 w Fs(The)401 b(pro)36 b(of)401 b(follo)-36 b(ws)403 b(the)d(same)h(metho)36 b(d)400 b(as)h(the)g(pro)36 b(of)401 b(of)g(Theorem)p 0 .5 0 TeXcolorrgb 39289 44376 a SDict begin H.S end 39289 44376 a 0 .5 0 TeXcolorrgb Fs(5)p 0 .5 0 TeXcolorrgb 39940 43520 a SDict begin H.R end 39940 43520 a 39940 44376 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5) cvn H.B /ANN pdfmark end 39940 44376 a Black Fs(.)567 b(F)-108 b(or)400 b(completeness,)408 b(w)-36 b(e)800 45981 y(giv)g(e)435 b(the)e(details.)578 b(W)-108 b(e)434 b(let)23124 47206 y(1)295 b(+)g Fq(y)p 20751 47799 7681 54 v 20751 49016 a Fs(1)g Fn(\000)h Fs(2)p Fq(xy)25096 48632 y Fo(2)25917 49016 y Fs(+)f Fq(y)27906 48632 y Fo(4)28933 48105 y Fs(=)369 b Fq(A)p Fs(\()p Fq(y)48 b Fs(\))800 50978 y(so)434 b(that)16760 52583 y Fq(A)p Fs(\()p Fq(y)48 b Fs(\))294 b Fn(\000)i Fs(2)p Fq(xy)23123 52035 y Fo(2)23649 52583 y Fq(A)p Fs(\()p Fq(y)48 b Fs(\))294 b(+)h Fq(y)28601 52035 y Fo(4)29127 52583 y Fq(A)p Fs(\()p Fq(y)48 b Fs(\))368 b(=)h(1)296 b(+)e Fq(y)48 b(:)800 54859 y Fs(Extracting)434 b(the)f(co)36 b(e\016cien)-36 b(t)434 b(of)g Fq(y)17758 54377 y Fp(n)p Fo(+4)19587 54859 y Fs(,)f(w)-36 b(e)434 b(obtain)11777 57679 y([)p Fq(y)12820 57131 y Fp(n)p Fo(+4)14649 57679 y Fs(])p Fq(A)p Fs(\()p Fq(y)48 b Fs(\))295 b Fn(\000)g Fs(2)p Fq(x)p Fs([)p Fq(y)21734 57131 y Fp(n)p Fo(+2)23563 57679 y Fs(])p Fq(A)p Fs(\()p Fq(y)48 b Fs(\))295 b(+)g([)p Fq(y)29238 57131 y Fp(n)29865 57679 y Fs(])p Fq(A)p Fs(\()p Fq(y)48 b Fs(\))368 b(=)h([)p Fq(y)35687 57131 y Fp(n)p Fo(+4)37516 57679 y Fs(]\(1)295 b(+)g Fq(y)48 b Fs(\))800 58894 y SDict begin H.S end 800 58894 a 800 58894 a SDict begin 14.5 H.A end 800 58894 a 800 58894 a SDict begin [ /View [/XYZ H.V] /Dest (equation.22) cvn H.B /DEST pdfmark end 800 58894 a 19508 62104 a Fq(A)20483 62303 y Fp(n)p Fo(+4)22606 62104 y Fn(\000)296 b Fs(2)p Fq(xA)26299 62303 y Fp(n)p Fo(+2)28423 62104 y Fs(+)f Fq(A)30705 62303 y Fp(n)31700 62104 y Fs(=)369 b(0)p Fq(:)16396 b Fs(\(22\))800 64380 y(By)434 b(direct)f(insp)36 b(ection,)434 b(the)f(\014rst)g(terms)g(of)h(the)f(sequence)h Fq(A)31701 64579 y Fp(n)32696 64380 y Fs(=)369 b Fq(a)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))434 b(are)17879 67200 y(1)p Fq(;)221 b Fs(1)p Fq(;)g Fs(2)p Fq(x;)g Fs(2)p Fq(x;)g Fs(4)p Fq(x)25674 66651 y Fo(2)26500 67200 y Fn(\000)295 b Fs(1)p Fq(;)221 b Fs(4)p Fq(x)30449 66651 y Fo(2)31272 67200 y Fn(\000)295 b Fs(1)p Fq(;)221 b(:)g(:)g(:)s(:)p Black 26150 70521 a Fs(22)p Black eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.23) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(W)-108 b(e)626 b(no)-36 b(w)626 b(set)f Fq(B)9344 -2473 y Fp(n)10667 -2672 y Fs(=)696 b Fq(A)13350 -2473 y Fp(n)13976 -2672 y Fq(A)14951 -2473 y Fp(n)p Fo(+1)17405 -2672 y Fs(and)626 b Fq(T)20889 -2473 y Fp(n)22211 -2672 y Fs(=)696 b Fq(A)24894 -2473 y Fp(n)25520 -2672 y Fq(A)26495 -2473 y Fp(n)p Fo(+3)28324 -2672 y Fs(,)674 b(and)625 b(try)h(to)g(determine)f(the)g(corresp)36 b(onding)800 -1067 y(generating)434 b(functions.)578 b(F)-108 b(rom)433 b(recurrence)g(\()p 0 .5 0 TeXcolorrgb 23479 -1067 a SDict begin H.S end 23479 -1067 a 0 .5 0 TeXcolorrgb Fs(22)p 0 .5 0 TeXcolorrgb 24779 -1923 a SDict begin H.R end 24779 -1923 a 24779 -1067 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.22) cvn H.B /ANN pdfmark end 24779 -1067 a Black Fs(\),)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)19295 1796 y Fq(A)20270 1995 y Fp(n)p Fo(+4)23205 1796 y Fs(=)1107 b(2)p Fq(xA)27688 1995 y Fp(n)p Fo(+2)29812 1796 y Fn(\000)295 b Fq(A)32115 1995 y Fp(n)19295 3733 y Fq(A)20270 3932 y Fp(n)p Fo(+5)23205 3733 y Fs(=)1107 b(2)p Fq(xA)27688 3932 y Fp(n)p Fo(+3)29812 3733 y Fn(\000)295 b Fq(A)32115 3932 y Fp(n)p Fo(+1)33944 3733 y Fq(:)800 6595 y Fs(Multiplying,)434 b(w)-36 b(e)434 b(obtain)8157 9458 y Fq(A)9132 9657 y Fp(n)p Fo(+4)10961 9458 y Fq(A)11936 9657 y Fp(n)p Fo(+5)14133 9458 y Fs(=)369 b(4)p Fq(x)16903 8909 y Fo(2)17429 9458 y Fq(A)18404 9657 y Fp(n)p Fo(+2)20233 9458 y Fq(A)21208 9657 y Fp(n)p Fo(+3)23332 9458 y Fn(\000)295 b Fs(2)p Fq(xA)27024 9657 y Fp(n)p Fo(+1)28853 9458 y Fq(A)29828 9657 y Fp(n)p Fo(+2)31952 9458 y Fn(\000)g Fs(2)p Fq(xA)35644 9657 y Fp(n)36271 9458 y Fq(A)37246 9657 y Fp(n)p Fo(+3)39370 9458 y Fs(+)g Fq(A)41652 9657 y Fp(n)42278 9458 y Fq(A)43253 9657 y Fp(n)p Fo(+1)45081 9458 y Fq(:)800 12320 y Fs(This)434 b(can)f(b)36 b(e)434 b(written)f(as:)14580 10715 y SDict begin H.S end 14580 10715 a 14580 10715 a SDict begin 14.5 H.A end 14580 10715 a 14580 10715 a SDict begin [ /View [/XYZ H.V] /Dest (equation.23) cvn H.B /DEST pdfmark end 14580 10715 a 15247 15182 a Fq(B)16235 15381 y Fp(n)p Fo(+4)18432 15182 y Fs(=)369 b(4)p Fq(x)21202 14634 y Fo(2)21728 15182 y Fq(B)22716 15381 y Fp(n)p Fo(+2)24840 15182 y Fn(\000)295 b Fs(2)p Fq(xB)28545 15381 y Fp(n)p Fo(+1)30670 15182 y Fn(\000)g Fs(2)p Fq(xT)34149 15381 y Fp(n)35071 15182 y Fs(+)g Fq(B)37366 15381 y Fp(n)37992 15182 y Fq(:)12135 b Fs(\(23\))800 18045 y(No)-36 b(w)434 b(m)-36 b(ultiplying)434 b(equation)g(\()p 0 .5 0 TeXcolorrgb 16516 18045 a SDict begin H.S end 16516 18045 a 0 .5 0 TeXcolorrgb Fs(22)p 0 .5 0 TeXcolorrgb 17817 17189 a SDict begin H.R end 17817 17189 a 17817 18045 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.22) cvn H.B /ANN pdfmark end 17817 18045 a Black Fs(\))f(b)-36 b(y)433 b Fq(A)21537 18244 y Fp(n)p Fo(+3)23366 18045 y Fs(,)h(w)-36 b(e)434 b(obtain)16428 20907 y Fq(A)17403 21106 y Fp(n)p Fo(+3)19232 20907 y Fq(A)20207 21106 y Fp(n)p Fo(+4)22405 20907 y Fs(=)368 b(2)p Fq(xA)26149 21106 y Fp(n)p Fo(+2)27978 20907 y Fq(A)28953 21106 y Fp(n)p Fo(+3)31077 20907 y Fn(\000)295 b Fq(A)33380 21106 y Fp(n)34007 20907 y Fq(A)34982 21106 y Fp(n)p Fo(+3)36810 20907 y Fq(;)800 23770 y Fs(or)2390 22164 y SDict begin H.S end 2390 22164 a 2390 22164 a SDict begin 14.5 H.A end 2390 22164 a 2390 22164 a SDict begin [ /View [/XYZ H.V] /Dest (equation.24) cvn H.B /DEST pdfmark end 2390 22164 a 20727 25375 a Fq(B)21715 25574 y Fp(n)p Fo(+3)23913 25375 y Fs(=)368 b(2)p Fq(xB)27670 25574 y Fp(n)p Fo(+2)29795 25375 y Fn(\000)295 b Fq(T)31885 25574 y Fp(n)32512 25375 y Fq(:)17615 b Fs(\(24\))800 27669 y(W)-108 b(e)374 b(can)f(no)-36 b(w)374 b(use)f(the)g(metho)36 b(d)373 b(of)h(co)36 b(e\016cien)-36 b(ts)374 b(to)g(pass)f(from)h(the)f(recurrences)g(to)g(the)g(corresp)36 b(onding)800 29274 y(generating)469 b(functions)f Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))468 b(and)g Fq(T)181 b Fs(\()p Fq(y)48 b Fs(\).)683 b(In)468 b(order)g(to)g(do)h(so,)478 b(w)-36 b(e)468 b(note)h(that)f(the)g(co)36 b(e\016cien)-36 b(ts)468 b(of)i(the)800 30879 y(sequence)434 b(generated)f(b)-36 b(y)433 b Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))433 b(b)36 b(egin:)19418 33741 y(1)p Fq(;)221 b Fs(2)p Fq(x;)g Fs(4)p Fq(x)24010 33193 y Fo(2)24538 33741 y Fq(;)g Fs(2)p Fq(x)p Fs(\(4)p Fq(x)28404 33193 y Fo(2)29227 33741 y Fn(\000)295 b Fs(1\))p Fq(;)221 b(:)g(:)g(:)s(:)800 36604 y Fs(Equation)434 b(\()p 0 .5 0 TeXcolorrgb 6924 36604 a SDict begin H.S end 6924 36604 a 0 .5 0 TeXcolorrgb Fs(23)p 0 .5 0 TeXcolorrgb 8224 35748 a SDict begin H.R end 8224 35748 a 8224 36604 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.23) cvn H.B /ANN pdfmark end 8224 36604 a Black Fs(\))g(b)36 b(ecomes)14338 34999 y SDict begin H.S end 14338 34999 a 14338 34999 a SDict begin 14.5 H.A end 14338 34999 a 14338 34999 a SDict begin [ /View [/XYZ H.V] /Dest (equation.25) cvn H.B /DEST pdfmark end 14338 34999 a 19072 39357 a Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))295 b Fn(\000)g Fs(1)h Fn(\000)f Fs(2)p Fq(xy)344 b Fn(\000)295 b Fs(4)p Fq(x)30802 38874 y Fo(2)31328 39357 y Fq(y)32010 38874 y Fo(2)32831 39357 y Fn(\000)g Fs(2)p Fq(x)p Fs(\(4)p Fq(x)37443 38874 y Fo(2)38265 39357 y Fn(\000)g Fs(1\))p Fq(y)41431 38874 y Fo(3)p 19072 39950 22886 54 v 29911 41166 a Fq(y)30593 40783 y Fo(4)11510 43762 y Fs(=)369 b(4)p Fq(x)14280 43214 y Fo(2)14939 42864 y Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))295 b Fn(\000)g Fs(1)g Fn(\000)h Fs(2)p Fq(xy)p 14939 43457 8718 54 v 18694 44674 a(y)19376 44290 y Fo(2)24084 43762 y Fn(\000)f Fs(2)p Fq(x)26934 42864 y(B)67 b Fs(\()p Fq(y)48 b Fs(\))296 b Fn(\000)f Fs(1)p 26934 43457 5023 54 v 29105 44674 a Fq(y)32385 43762 y Fn(\000)h Fs(2)p Fq(xT)181 b Fs(\()p Fq(y)48 b Fs(\))295 b(+)f Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))50488 41950 y(\(25\))800 47296 y(while)434 b(Equation)g(\()p 0 .5 0 TeXcolorrgb 10320 47296 a SDict begin H.S end 10320 47296 a 0 .5 0 TeXcolorrgb Fs(24)p 0 .5 0 TeXcolorrgb 11621 46440 a SDict begin H.R end 11621 46440 a 11621 47296 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.24) cvn H.B /ANN pdfmark end 11621 47296 a Black Fs(\))f(b)36 b(ecomes:)18240 45691 y SDict begin H.S end 18240 45691 a 18240 45691 a SDict begin 14.5 H.A end 18240 45691 a 18240 45691 a SDict begin [ /View [/XYZ H.V] /Dest (equation.26) cvn H.B /DEST pdfmark end 18240 45691 a 11696 50049 a Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))295 b Fn(\000)h Fs(1)f Fn(\000)h Fs(2)p Fq(xy)343 b Fn(\000)295 b Fs(4)p Fq(x)23426 49567 y Fo(2)23953 50049 y Fq(y)24635 49567 y Fo(2)p 11696 50642 13464 54 v 17825 51859 a Fq(y)18507 51475 y Fo(3)25662 50947 y Fs(=)369 b(2)p Fq(x)28565 50049 y(B)67 b Fs(\()p Fq(y)48 b Fs(\))295 b Fn(\000)g Fs(1)h Fn(\000)f Fs(2)p Fq(xy)p 28565 50642 8718 54 v 32320 51859 a(y)33002 51475 y Fo(2)37710 50947 y Fn(\000)h Fq(T)181 b Fs(\()p Fq(y)48 b Fs(\))p Fq(:)8451 b Fs(\(26\))800 54481 y(W)-108 b(e)496 b(no)-36 b(w)496 b(ha)-36 b(v)g(e)497 b(t)-36 b(w)g(o)496 b(linear)g(equations)h (for)f(the)g(t)-36 b(w)g(o)496 b(unkno)-36 b(wns)496 b Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))495 b(and)h Fq(T)181 b Fs(\()p Fq(y)48 b Fs(\).)765 b(Solving,)513 b(w)-36 b(e)496 b(\014nd)800 56086 y(that)18088 57952 y Fq(B)67 b Fs(\()p Fq(y)48 b Fs(\))369 b(=)28724 57053 y(1)p 22719 57646 12660 54 v 22719 58863 a(\(1)296 b Fn(\000)f Fq(y)26181 58479 y Fo(2)26707 58863 y Fs(\)\(1)g Fn(\000)g Fs(2)p Fq(xy)344 b Fs(+)294 b Fq(y)34347 58479 y Fo(2)34873 58863 y Fs(\))800 60917 y(as)434 b(required.)p 51860 60917 45 878 v 51905 60083 781 45 v 51905 60917 V 52684 60917 45 878 v 800 63498 a(W)-108 b(e)434 b(note)f(that)g(w)-36 b(e)434 b(also)h(obtain)17964 66621 y Fq(T)181 b Fs(\()p Fq(y)48 b Fs(\))368 b(=)26965 65722 y(2)p Fq(x)296 b Fn(\000)f Fq(y)p 22483 66315 12660 54 v 22483 67532 a Fs(\(1)g Fn(\000)g Fq(y)25944 67148 y Fo(2)26470 67532 y Fs(\)\(1)g Fn(\000)g Fs(2)p Fq(xy)344 b Fs(+)295 b Fq(y)34111 67148 y Fo(2)34636 67532 y Fs(\))35275 66621 y Fq(:)p Black 26150 70521 a Fs(23)p Black eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.24) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fs(This)434 b(theorem)f(then)g(implies)h (that)19024 -4277 y SDict begin H.S end 19024 -4277 a 19024 -4277 a SDict begin 14.5 H.A end 19024 -4277 a 19024 -4277 a SDict begin [ /View [/XYZ H.V] /Dest (equation.27) cvn H.B /DEST pdfmark end 19024 -4277 a 9874 1712 a Fq(w)36 b Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))369 b(=)15735 -177 y Fl(b)16286 -539 y Fj(n)p 16286 -378 512 40 v 16339 170 a Fk(2)16930 -177 y Fl(c)15582 450 y Fm(X)15683 3279 y Fp(k)24 b Fo(=0)17722 1712 y Fq(U)18605 1911 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)21169 -161 y Fm(\022)22279 814 y Fq(r)332 b Fs(+)295 b(2)p 22279 1407 2875 54 v 23391 2624 a(2)25287 -161 y Fm(\023)26634 1712 y Fs(=)368 b Fq(U)29036 1164 y Fp(d)28897 2041 y(n)29797 -161 y Fm(\022)30908 814 y Fq(r)331 b Fs(+)295 b(2)p 30908 1407 V 32020 2624 a(2)33915 -161 y Fm(\023)35115 1712 y Fq(U)36137 1164 y Fp(d)35998 2041 y(n)p Fo(+1)38048 -161 y Fm(\022)39158 814 y Fq(r)332 b Fs(+)295 b(2)p 39158 1407 V 40271 2624 a(2)42166 -161 y Fm(\023)43365 1712 y Fq(:)6762 b Fs(\(27\))800 5774 y(An)433 b(immediate)h(consequence)g(of)g(this)f(result)g(is)h (that)f(w)-36 b(e)434 b(ha)-36 b(v)g(e)7362 10295 y Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))1108 b(=)14287 8405 y Fl(b)14838 8043 y Fj(n)p 14838 8204 512 40 v 14891 8753 a Fk(2)15482 8405 y Fl(c)14134 9033 y Fm(X)14235 11862 y Fp(k)24 b Fo(=0)16274 10295 y Fq(U)17157 10494 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)19721 8422 y Fm(\022)20831 9396 y Fq(r)332 b Fs(+)295 b(2)p 20831 9989 2875 54 v 21944 11206 a(2)23839 8422 y Fm(\023)25112 10295 y Fn(\000)26440 8405 y Fl(b)26991 8032 y Fj(n)p Fi(\000)p Fk(1)p 26991 8204 1557 40 v 27567 8753 a(2)28681 8405 y Fl(c)26810 9033 y Fm(X)26912 11862 y Fp(k)24 b Fo(=0)29320 10295 y Fq(U)30203 10494 y Fp(n)p Fl(\000)p Fo(2)p Fp(k)g Fl(\000)p Fo(1)33969 8422 y Fm(\022)35080 9396 y Fq(r)332 b Fs(+)294 b(2)p 35080 9989 2875 54 v 36192 11206 a(2)38088 8422 y Fm(\023)12015 14364 y Fs(=)1107 b Fq(U)15156 13815 y Fp(d)15017 14692 y(n)15917 12491 y Fm(\022)17027 13465 y Fq(r)332 b Fs(+)295 b(2)p 17027 14058 V 18140 15275 a(2)20035 12491 y Fm(\023)21234 14364 y Fq(U)22256 13815 y Fp(d)22117 14692 y(n)p Fo(+1)24167 12491 y Fm(\022)25278 13465 y Fq(r)332 b Fs(+)294 b(2)p 25278 14058 V 26390 15275 a(2)28286 12491 y Fm(\023)29559 14364 y Fn(\000)h Fq(U)31909 13815 y Fp(d)31770 14692 y(n)p Fl(\000)p Fo(1)33820 12491 y Fm(\022)34931 13465 y Fq(r)331 b Fs(+)295 b(2)p 34931 14058 V 36043 15275 a(2)37938 12491 y Fm(\023)39138 14364 y Fq(U)40160 13815 y Fp(d)40021 14692 y(n)40921 12491 y Fm(\022)42031 13465 y Fq(r)332 b Fs(+)294 b(2)p 42031 14058 V 43143 15275 a(2)45039 12491 y Fm(\023)12015 17995 y Fs(=)1107 b Fq(U)15156 17446 y Fp(d)15017 18323 y(n)15917 16122 y Fm(\022)17027 17096 y Fq(r)332 b Fs(+)295 b(2)p 17027 17689 V 18140 18906 a(2)20035 16122 y Fm(\023)221 b(\022)22212 17995 y Fq(U)23234 17446 y Fp(d)23095 18323 y(n)p Fo(+1)25145 16122 y Fm(\022)26256 17096 y Fq(r)331 b Fs(+)295 b(2)p 26256 17689 V 27368 18906 a(2)29263 16122 y Fm(\023)30536 17995 y Fn(\000)h Fq(U)32887 17446 y Fp(d)32748 18323 y(n)p Fl(\000)p Fo(1)34798 16122 y Fm(\022)35909 17096 y Fq(r)331 b Fs(+)295 b(2)p 35909 17689 V 37021 18906 a(2)38916 16122 y Fm(\023\023)800 21692 y Fs(This)328 b(exhibits)g Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))328 b(as)g(the)f(pro)36 b(duct)326 b(of)j(a)f(Cheb)-36 b(yshev)327 b(related)h(p)36 b(olynomial)329 b(and)e(its)h(\014rst)e(di\013erence.)800 24026 y SDict begin H.S end 800 24026 a 800 24026 a SDict begin 14.5 H.A end 800 24026 a 800 24026 a SDict begin [ /View [/XYZ H.V] /Dest (section.8) cvn H.B /DEST pdfmark end 800 24026 a 2103 x Ft(8)2152 b(Additional)715 b(Riordan)h(arra)-60 b(y)717 b(factorizations)800 29049 y Fs(W)-108 b(e)434 b(note)f(that)14322 30213 y(1)295 b(+)g Fq(x)g Fn(\000)h Fq(x)19676 29731 y Fo(2)p 13748 30806 7029 54 v 13748 32023 a Fs(1)f Fn(\000)h Fq(r)36 b(x)17383 31639 y Fo(2)18204 32023 y Fs(+)295 b Fq(x)20250 31639 y Fo(4)21278 31111 y Fs(=)22658 29238 y Fm(\022)23769 30213 y Fs(1)g(+)g Fq(x)g Fn(\000)h Fq(x)29123 29731 y Fo(2)p 23769 30806 5880 54 v 24950 32023 a Fs(1)g(+)e Fq(x)27941 31639 y Fo(4)29782 31111 y Fq(;)31623 30213 y(x)32362 29731 y Fo(2)p 30497 30806 3518 54 v 30497 32023 a Fs(1)i(+)e Fq(x)33488 31639 y Fo(4)34147 29238 y Fm(\023)35420 31111 y Fn(\001)37710 30213 y Fs(1)p 36217 30806 3636 54 v 36217 32023 a(1)i Fn(\000)f Fq(r)36 b(x)800 34126 y Fs(and)433 b(hence)9400 36813 y(1)296 b(+)f Fq(x)g Fn(\000)g Fq(x)14754 36331 y Fo(2)p 7194 37406 10292 54 v 7194 38623 a Fs(1)h Fn(\000)f Fs(\()p Fq(r)331 b Fs(+)295 b(2\))p Fq(x)14093 38239 y Fo(2)14915 38623 y Fs(+)f Fq(x)16960 38239 y Fo(4)17988 37711 y Fs(=)19369 35838 y Fm(\022)20479 36813 y Fs(1)i(+)f Fq(x)g Fn(\000)g Fq(x)25833 36331 y Fo(2)p 20479 37406 5880 54 v 21661 38623 a Fs(1)g(+)g Fq(x)24652 38239 y Fo(4)26492 37711 y Fq(;)28334 36813 y(x)29073 36331 y Fo(2)p 27208 37406 3518 54 v 27208 38623 a Fs(1)g(+)g Fq(x)30199 38239 y Fo(4)30858 35838 y Fm(\023)221 b(\022)34349 36813 y Fs(1)p 33168 37406 3013 54 v 33168 38623 a(1)295 b Fn(\000)g Fq(x)36313 37711 y(;)38166 36813 y(x)p 37029 37406 V 37029 38623 a Fs(1)g Fn(\000)h Fq(x)40175 35838 y Fm(\023)41153 36136 y Fo(2)41974 37711 y Fn(\001)44263 36813 y Fs(1)p 42771 37406 3636 54 v 42771 38623 a(1)f Fn(\000)g Fq(r)36 b(x)800 41335 y Fs(Th)-36 b(us)7466 41719 y Fm(\022)8577 42693 y Fs(1)295 b(+)g Fq(x)g Fn(\000)h Fq(x)13931 42211 y Fo(2)p 8577 43286 5880 54 v 8979 44503 a Fs(\(1)f Fn(\000)g Fq(x)12497 44119 y Fo(2)13023 44503 y Fs(\))13529 44119 y Fo(2)14590 43592 y Fq(;)17211 42693 y(x)17950 42211 y Fo(2)p 15305 43286 5077 54 v 15305 44503 a Fs(\(1)g Fn(\000)h Fq(x)18824 44119 y Fo(2)19350 44503 y Fs(\))19856 44119 y Fo(2)20514 41719 y Fm(\023)21861 43592 y Fs(=)23242 41719 y Fm(\022)24352 42693 y Fs(1)g(+)f Fq(x)g Fn(\000)g Fq(x)29706 42211 y Fo(2)p 24352 43286 5880 54 v 25534 44503 a Fs(1)g(+)g Fq(x)28525 44119 y Fo(4)30365 43592 y Fq(;)32207 42693 y(x)32946 42211 y Fo(2)p 31080 43286 3518 54 v 31080 44503 a Fs(1)h(+)f Fq(x)34072 44119 y Fo(4)34731 41719 y Fm(\023)221 b(\022)38222 42693 y Fs(1)p 37040 43286 3013 54 v 37040 44503 a(1)296 b Fn(\000)f Fq(x)40186 43592 y(;)42039 42693 y(x)p 40902 43286 V 40902 44503 a Fs(1)g Fn(\000)h Fq(x)44048 41719 y Fm(\023)45025 42016 y Fo(2)45773 43592 y Fq(:)800 46813 y Fs(Here,)4232 45737 y Fm(\000)5604 46290 y Fo(1)p 4974 46508 1732 54 v 4974 47271 a(1)p Fl(\000)p Fp(x)6838 46813 y Fq(;)8155 46290 y Fp(x)p 7554 46508 V 7554 47271 a Fo(1)p Fl(\000)p Fp(x)9419 45737 y Fm(\001)10461 46813 y Fs(represen)-36 b(ts)432 b(the)h(standard)g(binomial)h (transformation.)579 b(W)-108 b(e)434 b(also)g(ha)-36 b(v)g(e)5812 50754 y(\(1)295 b(+)g Fq(x)g Fs(+)g Fq(x)11650 50205 y Fo(3)12471 50754 y Fn(\000)h Fq(x)14539 50205 y Fo(4)15065 50754 y Fq(;)221 b(x)16386 50205 y Fo(2)16912 50754 y Fs(\))295 b Fn(\001)18377 48881 y Fm(\022)19751 49855 y Fs(1)g Fn(\000)h Fq(x)p 19488 50448 3539 54 v 19488 51665 a Fs(1)g Fn(\000)f Fq(x)22501 51281 y Fo(4)23160 50754 y Fq(;)25264 49855 y(x)p 23875 50448 3518 54 v 23875 51665 a Fs(1)h(+)f Fq(x)26867 51281 y Fo(2)27525 48881 y Fm(\023)29610 50754 y Fs(=)31729 48881 y Fm(\022)33091 49855 y Fs(1)h(+)f Fq(x)g Fs(+)g Fq(x)38424 49373 y Fo(3)39245 49855 y Fn(\000)g Fq(x)41312 49373 y Fo(4)p 32839 50448 9252 54 v 32839 51665 a Fs(1)h(+)f Fq(x)35831 51281 y Fo(2)36652 51665 y Fs(+)g Fq(x)38698 51281 y Fo(4)39519 51665 y Fs(+)g Fq(x)41565 51281 y Fo(6)42223 50754 y Fq(;)44065 49855 y(x)44804 49373 y Fo(2)p 42939 50448 3518 54 v 42939 51665 a Fs(1)g(+)g Fq(x)45930 51281 y Fo(4)46589 48881 y Fm(\023)29610 54410 y Fs(=)31729 52537 y Fm(\022)32839 53511 y Fs(1)h(+)f Fq(x)g Fn(\000)g Fq(x)38193 53029 y Fo(2)p 32839 54105 5880 54 v 34021 55321 a Fs(1)g(+)g Fq(x)37012 54937 y Fo(4)38852 54410 y Fq(;)40694 53511 y(x)41433 53029 y Fo(2)p 39568 54105 3518 54 v 39568 55321 a Fs(1)g(+)g Fq(x)42559 54937 y Fo(4)43218 52537 y Fm(\023)44417 54410 y Fq(:)800 58034 y Fs(A)434 b(consequence)f(of)h (this)g(last)g(factorization)h(is)f(the)f(fact)h(that)7390 61682 y(\(1)296 b(+)e Fq(x)i Fs(+)f Fq(x)13229 61134 y Fo(3)14050 61682 y Fn(\000)g Fq(x)16117 61134 y Fo(4)16643 61682 y Fq(;)221 b(x)17964 61134 y Fo(2)18491 61682 y Fs(\))295 b Fn(\001)28055 60784 y Fs(1)p 20089 61377 16584 54 v 20089 62593 a(1)g Fn(\000)g Fs(\()p Fq(r)332 b Fn(\000)295 b Fs(1\))p Fq(x)g Fn(\000)h Fs(\()p Fq(r)331 b Fn(\000)295 b Fs(1\))p Fq(x)33279 62210 y Fo(2)34101 62593 y Fs(+)g Fq(x)36147 62210 y Fo(3)37174 61682 y Fs(=)39262 60784 y(1)g(+)g Fq(x)g Fn(\000)h Fq(x)44616 60302 y Fo(2)p 38688 61377 7029 54 v 38688 62593 a Fs(1)f Fn(\000)h Fq(r)36 b(x)42323 62210 y Fo(2)43144 62593 y Fs(+)295 b Fq(x)45190 62210 y Fo(4)45849 61682 y Fq(:)p Black 26150 70521 a Fs(24)p Black eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.25) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 800 -4000 a SDict begin H.S end 800 -4000 a 800 -4000 a SDict begin 14.5 H.A end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (section.9) cvn H.B /DEST pdfmark end 800 -4000 a 1328 x Ft(9)2152 b(A)717 b(left)g(in)-60 b(v)g(erse)800 249 y Fs(W)-108 b(e)415 b(no)-36 b(w)415 b(seek)h(a)g(left)f(in)-36 b(v)g(erse)416 b([)p 0 .5 0 TeXcolorrgb 16456 249 a SDict begin H.S end 16456 249 a 0 .5 0 TeXcolorrgb Fs(1)p 0 .5 0 TeXcolorrgb 17106 -607 a SDict begin H.R end 17106 -607 a 17106 249 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.CMS) cvn H.B /ANN pdfmark end 17106 249 a Black Fs(])g(for)f(the)g(arra)-36 b(y)416 b(\(1)257 b(+)g Fq(x)h Fs(+)f Fq(x)31138 -233 y Fo(3)31922 249 y Fn(\000)g Fq(x)33951 -233 y Fo(4)34477 249 y Fq(;)221 b(x)35798 -233 y Fo(2)36325 249 y Fs(\).)572 b(T)-108 b(o)415 b(this)g(end)g(w)-36 b(e)415 b(let)g Fq(t)369 b Fs(=)50593 -708 y Fn(p)p 51700 -708 740 54 v 957 x Fq(x)p Fs(.)800 1854 y(Then)12107 5519 y([)p Fq(x)13207 4971 y Fp(n)13834 5519 y Fs(])19155 3664 y Fn(p)p 20261 3664 V 20261 4621 a Fq(x)21001 3908 y Fp(k)p 14328 5214 12068 54 v 14328 6656 a Fs(1)296 b(+)16580 5699 y Fn(p)p 17687 5699 740 54 v 957 x Fq(x)g Fs(+)20029 5699 y Fn(p)p 21135 5699 V 21135 6656 a Fq(x)21875 5944 y Fo(3)22696 6656 y Fn(\000)24024 5699 y(p)p 25131 5699 V 957 x Fq(x)25870 5944 y Fo(4)27636 5519 y Fs(=)1106 b([)p Fq(t)30585 4971 y Fo(2)p Fp(n)31682 5519 y Fs(])35626 4621 y Fq(t)36096 4138 y Fp(k)p 32176 5214 7939 54 v 32176 6430 a Fs(1)296 b(+)e Fq(t)h Fs(+)g Fq(t)36970 6047 y Fo(3)37791 6430 y Fn(\000)g Fq(t)39589 6047 y Fo(4)27636 9172 y Fs(=)1106 b([)p Fq(t)30585 8624 y Fo(2)p Fp(n)p Fl(\000)p Fp(k)32927 9172 y Fs(])37066 8273 y(1)p 33421 8867 V 33421 10083 a(1)296 b(+)f Fq(t)g Fs(+)f Fq(t)38215 9700 y Fo(3)39036 10083 y Fn(\000)h Fq(t)40834 9700 y Fo(4)27636 12811 y Fs(=)1106 b([)p Fq(t)30585 12263 y Fo(2)p Fp(n)p Fl(\000)p Fp(k)32927 12811 y Fs(])33999 11151 y Fl(1)33510 11549 y Fm(X)33652 14349 y Fp(j)51 b Fo(=0)35650 12811 y Fq(u)36390 13010 y Fp(j)36877 12811 y Fq(x)37616 12263 y Fp(j)27636 15669 y Fs(=)1106 b Fq(u)30494 15868 y Fo(2)p Fp(n)p Fl(\000)p Fp(k)800 18602 y Fs(where)16891 20467 y Fq(u)17631 20666 y Fp(n)18627 20467 y Fs(=)368 b(\()p Fn(\000)p Fs(1\))22702 19919 y Fp(n)23329 20467 y Fq(F)181 b Fs(\()p Fn(b)25581 19569 y Fq(n)295 b Fs(+)g(2)p 25581 20162 3029 54 v 26770 21379 a(2)28742 20467 y Fn(c)p Fs(\))p Fq(F)181 b Fs(\()p Fn(b)32090 19569 y Fq(n)295 b Fs(+)g(3)p 32090 20162 V 33279 21379 a(2)35251 20467 y Fn(c)p Fs(\))p Fq(:)800 23131 y Fs(W)-108 b(e)434 b(deduce)e(from)i (this)f(that)5853 27416 y SDict begin H.S end 5853 27416 a 5853 27416 a SDict begin 14.5 H.A end 5853 27416 a 5853 27416 a SDict begin [ /View [/XYZ H.V] /Dest (equation.28) cvn H.B /DEST pdfmark end 5853 27416 a Fq(b)p Fs(\()p Fq(n)p Fs(;)221 b Fq(r)36 b Fs(\))1108 b(=)13063 25756 y Fo(2)p Fp(n)12625 26154 y Fm(X)12726 28983 y Fp(k)24 b Fo(=0)14543 27416 y Fs(\()p Fn(\000)p Fs(1\))17238 26868 y Fp(k)17807 27416 y Fq(F)181 b Fs(\()p Fn(b)20059 26517 y Fs(2)p Fq(n)296 b Fn(\000)f Fq(k)341 b Fs(+)294 b(2)p 20059 27111 6024 54 v 22746 28327 a(2)26216 27416 y Fn(c)p Fs(\))p Fq(F)181 b Fs(\()p Fn(b)29564 26517 y Fs(2)p Fq(n)295 b Fn(\000)h Fq(k)340 b Fs(+)295 b(3)p 29564 27111 V 32250 28327 a(2)35720 27416 y Fn(c)p Fs(\))37219 25526 y Fl(b)37770 25164 y Fj(k)p 37771 25325 453 40 v 37794 25874 a Fk(2)38356 25526 y Fl(c)37038 26154 y Fm(X)37180 28954 y Fp(j)51 b Fo(=0)39178 25543 y Fm(\022)40155 26517 y Fq(k)341 b 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a 0 1 0 0 TeXcolorcmyk -266 x Fs(A002530)p 24563 45171 4878 54 v 0 1 0 0 TeXcolorcmyk 29440 44051 a SDict begin H.R end 29440 44051 a 29440 44959 a SDict begin [ /H /I /Border [0 0 0] /Color [0 1 1] /Action << /Subtype /URI /URI (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002530) >> /Subtype /Link H.B /ANN pdfmark end 29440 44959 a Black Fs(\),)e(for)h(example,)4556 50768 y Fm(0)4556 53106 y(B)4556 53903 y(B)4556 54700 y(B)4556 55497 y(B)4556 56294 y(B)4556 57091 y(B)4556 57888 y(B)4556 58685 y(B)4556 59482 y(B)4556 60332 y(@)6597 51921 y Fs(1)2274 b(0)g(0)g(0)1949 b(0)1624 b(0)g(0)g(0)g(0)g(0)f(0)221 b Fq(:)g(:)g(:)6597 53526 y Fs(1)1757 b Fn(\000)p Fs(1)h(1)2274 b(0)1949 b(0)1624 b(0)g(0)g(0)g(0)g(0)f(0)221 b Fq(:)g(:)g(:)6597 55131 y Fs(4)1757 b Fn(\000)p Fs(2)h(1)g Fn(\000)p Fs(1)1432 b(1)1624 b(0)g(0)g(0)g(0)g(0)f(0)221 b Fq(:)g(:)g(:)6597 56736 y Fs(9)1757 b Fn(\000)p Fs(6)h(4)g Fn(\000)p Fs(2)1432 b(1)1108 b Fn(\000)p Fs(1)f(1)1624 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Fm(0)43079 53106 y(B)43079 53903 y(B)43079 54700 y(B)43079 55497 y(B)43079 56294 y(B)43079 57091 y(B)43079 57888 y(B)43079 58685 y(B)43079 59482 y(B)43079 60332 y(@)45445 51921 y Fs(1)45445 53526 y(3)45120 55131 y(12)45120 56736 y(44)44795 58341 y(165)44795 59946 y(615)45590 61221 y(.)45590 61664 y(.)45590 62107 y(.)47299 50768 y Fm(1)47299 53106 y(C)47299 53903 y(C)47299 54700 y(C)47299 55497 y(C)47299 56294 y(C)47299 57091 y(C)47299 57888 y(C)47299 58685 y(C)47299 59482 y(C)47299 60332 y(A)48683 57025 y Fq(:)p Black 26150 70521 a Fs(25)p Black eop end end %%Page: 26 26 TeXDict begin HPSdict begin 26 25 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin 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