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0 TeXcolorrgb 8847 26102 a SDict begin H.S end 8847 26102 a 0 .5 0 TeXcolorrgb Fk(6)p 0 .5 0 TeXcolorrgb 9497 25246 a SDict begin H.R end 9497 25246 a 9497 26102 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 9497 26102 a Black Fk(\),)406 b(\()p 0 .5 0 TeXcolorrgb 11276 26102 a SDict begin H.S end 11276 26102 a 0 .5 0 TeXcolorrgb Fk(7)p 0 .5 0 TeXcolorrgb 11926 25246 a SDict begin H.R end 11926 25246 a 11926 26102 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 11926 26102 a Black Fk(\))398 b(are)h(kno)-36 b(wn)398 b(and)g(can)g(b)36 b(e)398 b(found,)406 b(with)398 b(pro)36 b(ofs)399 b(based)f(on)g (\014nite)g(di\013erences,)405 b(in)800 27707 y(the)467 b(C.)h(Jordan's)g(b)36 b(o)g(ok)469 b([)p 0 .5 0 TeXcolorrgb 13715 27707 a SDict begin H.S end 13715 27707 a 0 .5 0 TeXcolorrgb Fk(6)p 0 .5 0 TeXcolorrgb 14366 26851 a SDict begin H.R end 14366 26851 a 14366 27707 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.J) cvn H.B /ANN pdfmark end 14366 27707 a Black Fk(,)476 b(pp.)467 b(205{206].)683 b(The)468 b(form)-36 b(ulas)468 b(are)g(giv)-36 b(en,)477 b(as)468 b(exercises)h(without)e(pro)36 b(ofs)800 29312 y(and)338 b(without)h(direct)f(referencing)h(to)g([)p 0 .5 0 TeXcolorrgb 19944 29312 a SDict begin H.S end 19944 29312 a 0 .5 0 TeXcolorrgb Fk(6)p 0 .5 0 TeXcolorrgb 20595 28456 a SDict begin H.R end 20595 28456 a 20595 29312 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.J) cvn H.B /ANN pdfmark end 20595 29312 a Black Fk(],)358 b(also)340 b(in)f(the)f(L.)h(Com)-36 b(tet's)339 b(b)36 b(o)g(ok)340 b([)p 0 .5 0 TeXcolorrgb 38192 29312 a SDict begin H.S end 38192 29312 a 0 .5 0 TeXcolorrgb Fk(2)p 0 .5 0 TeXcolorrgb 38843 28456 a SDict begin H.R end 38843 28456 a 38843 29312 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.C) cvn H.B /ANN pdfmark end 38843 29312 a Black Fk(,)358 b(Ex.)339 b(6,)358 b(p.)339 b(157].)548 b(It)339 b(is)g(easy)800 30917 y(to)380 b(see)f(that)g(form)-36 b(ula)381 b(\()p 0 .5 0 TeXcolorrgb 12370 30917 a SDict begin H.S end 12370 30917 a 0 .5 0 TeXcolorrgb Fk(6)p 0 .5 0 TeXcolorrgb 13020 30061 a SDict begin H.R end 13020 30061 a 13020 30917 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 13020 30917 a Black Fk(\))e(holds,)390 b(with)380 b(the)f(same)h(pro)36 b(of,)391 b(for)380 b(complex)g(v)-72 b(ariable)381 b Fh(t)e Fk(and)g(a)h(holomorphic)800 32522 y(function)607 b Fh(f)142 b Fk(.)1097 b(F)-108 b(orm)-36 b(ula)607 b(\()p 0 .5 0 TeXcolorrgb 14101 32522 a SDict begin H.S end 14101 32522 a 0 .5 0 TeXcolorrgb Fk(7)p 0 .5 0 TeXcolorrgb 14751 31666 a SDict begin H.R end 14751 31666 a 14751 32522 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 14751 32522 a Black Fk(\))f(holds)h(for)g(complex)h Fh(t)p Fk(,)650 b(a)607 b(branc)-36 b(h)606 b(of)h(logarithm)h(and)e(a)h(holomorphic) 800 34128 y(function)433 b Fh(f)142 b Fk(.)800 36363 y(F)-108 b(or)433 b(example)h(if)h Fh(f)142 b Fk(\()p Fh(x)p Fk(\))368 b(=)h Fh(x)14439 35881 y Fi(m)15327 36363 y Fh(;)590 b(g)48 b Fk(\()p Fh(t)p Fk(\))368 b(=)h Fh(e)20783 35881 y Fi(mt)22444 36363 y Fk(then)432 b(using)i(\()p 0 .5 0 TeXcolorrgb 29316 36363 a SDict begin H.S end 29316 36363 a 0 .5 0 TeXcolorrgb Fk(6)p 0 .5 0 TeXcolorrgb 29966 35507 a SDict begin H.R end 29966 35507 a 29966 36363 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 29966 36363 a Black Fk(\))f(w)-36 b(e)434 b(get)34988 34757 y SDict begin H.S end 34988 34757 a 34988 34757 a SDict begin 14.5 H.A end 34988 34757 a 34988 34757 a SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark end 34988 34757 a 11726 40135 a Fh(g)12397 39586 y Ff(\()p Fi(n)p Ff(\))13754 40135 y Fk(\()p Fh(t)p Fk(\))369 b(=)17659 38474 y Fi(n)16985 38873 y Fj(X)17087 41702 y Fi(k)24 b Ff(=1)19125 38262 y Fj(\032)20122 39236 y Fh(n)20149 41046 y(k)20898 38262 y Fj(\033)21894 40135 y Fh(m)p Fk(\()p Fh(m)295 b Fe(\000)g Fk(1\))221 b Fe(\001)g(\001)g(\001)i Fk(\()p Fh(m)294 b Fe(\000)i Fh(k)340 b Fk(+)295 b(1\))p Fh(e)36797 39586 y Ff(\()p Fi(m)p Fd(\000)p Fi(k)24 b Ff(\))p Fi(t)40001 40135 y Fh(e)40604 39586 y Fi(k)g(t)41513 40135 y Fh(:)9264 b Fk(\(8\))800 44022 y(F)-108 b(rom)433 b(the)g(other)g(side)12459 42417 y SDict begin H.S end 12459 42417 a 12459 42417 a SDict begin 14.5 H.A end 12459 42417 a 12459 42417 a SDict begin [ /View [/XYZ H.V] /Dest (equation.9) cvn H.B /DEST pdfmark end 12459 42417 a 22193 45627 a Fh(g)22864 45079 y Ff(\()p Fi(n)p Ff(\))24221 45627 y Fk(\()p Fh(t)p Fk(\))368 b(=)h Fh(m)28590 45079 y Fi(n)29216 45627 y Fh(e)29819 45079 y Fi(mt)31046 45627 y Fh(;)19731 b Fk(\(9\))800 47877 y(and)520 b(comparing)h(\()p 0 .5 0 TeXcolorrgb 10368 47877 a SDict begin H.S end 10368 47877 a 0 .5 0 TeXcolorrgb Fk(8)p 0 .5 0 TeXcolorrgb 11018 47021 a SDict begin H.R end 11018 47021 a 11018 47877 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.8) cvn H.B /ANN pdfmark end 11018 47877 a Black Fk(\))f(with)h(\()p 0 .5 0 TeXcolorrgb 15600 47877 a SDict begin H.S end 15600 47877 a 0 .5 0 TeXcolorrgb Fk(9)p 0 .5 0 TeXcolorrgb 16250 47021 a SDict begin H.R end 16250 47021 a 16250 47877 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.9) cvn H.B /ANN pdfmark end 16250 47877 a Black Fk(\))f(w)-36 b(e)521 b(obtain)f(the)g(w)-36 b(ell-kno)g(wn)521 b(generating)g(function)f (for)h(the)f(Stirling)800 49483 y(n)-36 b(um)g(b)36 b(ers)432 b(of)i(the)f(second)h(kind)21723 50107 y Fi(n)21049 50506 y Fj(X)21150 53335 y Fi(k)24 b Ff(=1)23189 49895 y Fj(\032)24185 50869 y Fh(n)24213 52679 y(k)24962 49895 y Fj(\033)25958 51768 y Fk(\()p Fh(m)p Fk(\))28108 51967 y Fi(k)29045 51768 y Fk(=)369 b Fh(m)31564 51219 y Fi(n)32190 51768 y Fh(:)800 55444 y SDict begin H.S end 800 55444 a 800 55444 a SDict begin 14.5 H.A end 800 55444 a 800 55444 a SDict begin [ /View [/XYZ H.V] /Dest (section.3) cvn H.B /DEST pdfmark end 800 55444 a 1439 x Fl(3)2152 b(Addition)540 b(form)-60 b(ulas)542 b(for)g(Stirling)f(n)-60 b(um)g(b)60 b(ers)541 b(of)h(the)g(second)4028 59318 y(kind)800 62239 y Fk(Let)433 b(us)g(substitute)g Fh(f)142 b Fk(\()p Fh(x)p Fk(\))368 b(=)h(exp\()p Fh(x)p Fk(\))433 b(in)h(\()p 0 .5 0 TeXcolorrgb 21362 62239 a SDict begin H.S end 21362 62239 a 0 .5 0 TeXcolorrgb Fk(6)p 0 .5 0 TeXcolorrgb 22012 61383 a SDict begin H.R end 22012 61383 a 22012 62239 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.6) cvn H.B /ANN pdfmark end 22012 62239 a Black Fk(\).)578 b(W)-108 b(e)434 b(ha)-36 b(v)g(e)28695 60633 y SDict begin H.S end 28695 60633 a 28695 60633 a SDict begin 14.5 H.A end 28695 60633 a 28695 60633 a SDict begin [ /View [/XYZ H.V] /Dest (equation.10) cvn H.B /DEST pdfmark end 28695 60633 a 17684 66164 a Fk(\()p Fh(e)18793 65616 y Fi(e)19229 65303 y Fb(t)19657 66164 y Fk(\))20163 65616 y Ff(\()p Fi(u)p Ff(+)p Fi(v)32 b Ff(\))23085 66164 y Fk(=)24542 64504 y Fi(u)p Ff(+)p Fi(v)24466 64902 y Fj(X)24567 67731 y Fi(k)24 b Ff(=1)26606 64291 y Fj(\032)27602 65266 y Fh(u)296 b Fk(+)e Fh(v)28751 67076 y(k)30621 64291 y Fj(\033)31617 66164 y Fh(e)32220 65616 y Fi(e)32656 65303 y Fb(t)33379 66164 y Fe(\001)h Fh(e)34646 65616 y Fi(k)24 b(t)35555 66164 y Fh(:)14572 b Fk(\(10\))p Black 26475 70521 a(3)p Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 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.4) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1328 x Fk(F)-108 b(rom)433 b(the)g(other)g(side) 19568 -386 y(\()p Fh(e)20677 -935 y Fi(e)21113 -1247 y Fb(t)21541 -386 y Fk(\))22047 -935 y Ff(\()p Fi(u)p Ff(\))23747 -386 y Fk(=)25815 -2047 y Fi(u)25128 -1648 y Fj(X)25201 1141 y Fi(n)p Ff(=1)27268 -2259 y Fj(\032)28282 -1285 y Fh(u)28264 525 y(n)29041 -2259 y Fj(\033)30037 -386 y Fh(e)30640 -935 y Fi(e)31076 -1247 y Fb(t)31799 -386 y Fe(\001)295 b Fh(e)33066 -935 y Fi(nt)800 3027 y Fk(and)12043 6640 y SDict begin H.S end 12043 6640 a 12043 6640 a SDict begin 14.5 H.A end 12043 6640 a 12043 6640 a SDict begin [ /View [/XYZ H.V] /Dest (equation.11) cvn H.B /DEST pdfmark end 12043 6640 a Fk(\()p Fh(e)13152 6092 y Fi(e)13588 5779 y Fb(t)14015 6640 y Fk(\))14521 6092 y Ff(\()p Fi(u)p Ff(+)p Fi(v)32 b Ff(\))17444 6640 y Fk(=)368 b([\()p Fh(e)20294 6092 y Fi(e)20730 5779 y Fb(t)21158 6640 y Fk(\))21664 6092 y Ff(\()p Fi(u)p Ff(\))22996 6640 y Fk(])23357 6092 y Ff(\()p Fi(v)32 b Ff(\))25003 6640 y Fk(=)27071 4980 y Fi(u)26384 5378 y Fj(X)26457 8168 y Fi(n)p Ff(=1)28524 4767 y Fj(\032)29538 5742 y Fh(u)29520 7552 y(n)30297 4767 y Fj(\033)31293 6640 y Fk(\()p Fh(e)32402 6092 y Fi(e)32838 5779 y Fb(t)33561 6640 y Fe(\001)295 b Fh(e)34828 6092 y Fi(nt)35794 6640 y Fk(\))36300 6092 y Ff(\()p Fi(v)32 b Ff(\))14257 10918 y Fk(=)16324 9258 y Fi(u)15637 9656 y Fj(X)15710 12446 y Fi(n)p Ff(=1)17777 9045 y Fj(\032)18792 10020 y Fh(u)18774 11829 y(n)19550 9045 y Fj(\033)21540 9258 y Fi(v)20826 9656 y Fj(X)20768 12446 y Fi(m)p Ff(=0)23024 9045 y Fj(\022)24232 10020 y Fh(v)24001 11829 y(m)25139 9045 y Fj(\023)26117 10918 y Fk(\()27166 9258 y Fi(m)26623 9656 y Fj(X)26821 12456 y Fi(i)p Ff(=0)28763 9045 y Fj(\032)29759 10020 y Fh(m)30106 11829 y(i)30897 9045 y Fj(\033)31893 10918 y Fh(e)32496 10370 y Fi(e)32932 10057 y Fb(t)33655 10918 y Fe(\001)295 b Fh(e)34922 10370 y Fi(it)35638 10918 y Fk(\))g Fe(\001)g Fh(n)37879 10370 y Fi(v)32 b Fd(\000)p Fi(m)39989 10918 y Fh(e)40592 10370 y Fi(nt)14257 15206 y Fk(=)16324 13545 y Fi(u)15637 13944 y Fj(X)15710 16733 y Fi(n)p Ff(=1)17777 13333 y Fj(\032)18792 14307 y Fh(u)18774 16117 y(n)19550 13333 y Fj(\033)21540 13545 y Fi(v)20826 13944 y Fj(X)20768 16733 y Fi(m)p Ff(=0)23024 13333 y Fj(\022)24232 14307 y Fh(v)24001 16117 y(m)25139 13333 y Fj(\023)26117 15206 y Fk(\()27166 13545 y Fi(m)26623 13944 y Fj(X)26821 16743 y Fi(i)p Ff(=0)28763 13333 y Fj(\032)29759 14307 y Fh(m)30106 16117 y(i)30897 13333 y Fj(\033)31893 15206 y Fh(e)32496 14657 y Fi(e)32932 14345 y Fb(t)33655 15206 y Fe(\001)295 b Fh(e)34922 14657 y Ff(\()p Fi(i)p Ff(+)p Fi(n)p Ff(\))p Fi(t)37672 15206 y Fh(n)38448 14657 y Fi(v)32 b Fd(\000)p Fi(m)40558 15206 y Fk(\))p Fh(:)p Black 9063 w Fk(\(11\))p Black 800 18204 a SDict begin H.S end 800 18204 a 800 18204 a SDict begin 14.5 H.A end 800 18204 a 800 18204 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2) cvn H.B /DEST pdfmark end 800 18204 a Black 1034 x Fc(Theorem)660 b(2.)p Black 625 w Fg(The)592 b(fol)66 b(lowing)593 b(addition)g(formula)g(for)f(the)h (Stirling)e(numb)-66 b(ers)592 b(of)h(the)g(se)-66 b(c)g(ond)592 b(kind)800 20843 y(holds.)4652 19238 y SDict begin H.S end 4652 19238 a 4652 19238 a SDict begin 14.5 H.A end 4652 19238 a 4652 19238 a SDict begin [ /View [/XYZ H.V] /Dest (equation.12) cvn H.B /DEST pdfmark end 4652 19238 a 13548 21489 a Fj(\032)14544 22463 y Fh(u)74 b Fk(+)g Fh(v)15472 24273 y(k)17120 21489 y Fj(\033)18485 23362 y Fk(=)20569 21701 y Fi(k)19866 22100 y Fj(X)19939 24889 y Fi(n)p Ff(=1)22006 21489 y Fj(\032)23020 22463 y Fh(u)23002 24273 y(n)23779 21489 y Fj(\033)26442 21701 y Fi(v)25727 22100 y Fj(X)24996 24929 y Fi(m)p Ff(=)p Fi(k)24 b Fd(\000)p Fi(n)28598 21489 y Fj(\022)29807 22463 y Fh(v)29576 24273 y(m)30714 21489 y Fj(\023\032)33679 22463 y Fh(m)32688 24273 y(k)340 b Fe(\000)295 b Fh(n)35809 21489 y Fj(\033)36805 23362 y Fh(n)37581 22813 y Fi(v)32 b Fd(\000)p Fi(m)39691 23362 y Fh(:)10436 b Fk(\(12\))p Black 800 27574 a Fg(Pr)-66 b(o)g(of.)p Black 649 w Fk(F)-108 b(orm)-36 b(ula)651 b(\()p 0 .5 0 TeXcolorrgb 10708 27574 a SDict begin H.S end 10708 27574 a 0 .5 0 TeXcolorrgb Fk(12)p 0 .5 0 TeXcolorrgb 12008 26718 a SDict begin H.R end 12008 26718 a 12008 27574 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.12) cvn H.B /ANN pdfmark end 12008 27574 a Black Fk(\))g(follo)-36 b(ws)653 b(b)-36 b(y)651 b(comparing)g(the)f(co)36 b(e\016cien)-36 b(ts)652 b(of)f Fh(e)37812 27092 y Fi(e)38248 26780 y Fb(t)38676 27574 y Fh(e)39279 27092 y Fi(k)24 b(t)40839 27574 y Fk(in)650 b(\()p 0 .5 0 TeXcolorrgb 43079 27574 a SDict begin H.S end 43079 27574 a 0 .5 0 TeXcolorrgb Fk(10)p 0 .5 0 TeXcolorrgb 44380 26718 a SDict begin H.R end 44380 26718 a 44380 27574 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.10) cvn H.B /ANN pdfmark end 44380 27574 a Black Fk(\))h(and)f(\()p 0 .5 0 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end 35482 38461 a 35482 38461 a SDict begin [ /View [/XYZ H.V] /Dest (equation.13) cvn H.B /DEST pdfmark end 35482 38461 a 12989 42397 a Fj(\032)13985 43371 y Fh(u)74 b Fk(+)g Fh(v)14912 45181 y(k)16561 42397 y Fj(\033)17926 44270 y Fk(=)19408 42609 y Fi(k)24 b Fd(\000)p Ff(1)19307 43008 y Fj(X)19449 45807 y Fi(j)51 b Ff(=0)21580 42555 y Fi(v)32 b Fd(\000)p Fi(j)21447 43008 y Fj(X)21645 45807 y Fi(i)p Ff(=0)23587 42397 y Fj(\022)24565 43371 y Fh(v)24681 45181 y(i)25241 42397 y Fj(\023)26219 44270 y Fk(\()p Fh(k)340 b Fe(\000)296 b Fh(j)75 b Fk(\))30189 43721 y Fi(i)30564 42397 y Fj(\032)31560 43371 y Fh(v)343 b Fe(\000)295 b Fh(i)32626 45181 y(j)34304 42397 y Fj(\033\032)37405 43371 y Fh(u)36296 45181 y(k)341 b Fe(\000)295 b Fh(j)39254 42397 y Fj(\033)40250 44270 y Fh(:)9877 b Fk(\(13\))800 48548 y(F)-108 b(orm)-36 b(ula)348 b(\()p 0 .5 0 TeXcolorrgb 6333 48548 a SDict begin H.S end 6333 48548 a 0 .5 0 TeXcolorrgb Fk(4)p 0 .5 0 TeXcolorrgb 6983 47692 a SDict begin H.R end 6983 47692 a 6983 48548 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4) cvn H.B /ANN pdfmark end 6983 48548 a Black Fk(\))g(follo)-36 b(ws)350 b(easily)f(from)g(\()p 0 .5 0 TeXcolorrgb 19012 48548 a SDict begin H.S end 19012 48548 a 0 .5 0 TeXcolorrgb Fk(13)p 0 .5 0 TeXcolorrgb 20312 47692 a SDict begin H.R end 20312 47692 a 20312 48548 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 20312 48548 a Black Fk(\).)550 b(T)-108 b(o)349 b(see)f(this)g(let)g(us)g(observ)-36 b(e)348 b(that)g(the)f(range)i(of)f(the)g(v)-72 b(ariables)800 50153 y Fh(i;)221 b(j)427 b Fk(in)353 b(\()p 0 .5 0 TeXcolorrgb 4734 50153 a SDict begin H.S end 4734 50153 a 0 .5 0 TeXcolorrgb Fk(13)p 0 .5 0 TeXcolorrgb 6034 49297 a SDict begin H.R end 6034 49297 a 6034 50153 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.13) cvn H.B /ANN pdfmark end 6034 50153 a Black Fk(\))f(can)g(b)36 b(e)352 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0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.21) cvn H.B /ANN pdfmark end 8107 37699 a Black Fk(\))g(w)-36 b(e)423 b(obtain)f(the)f(follo)-36 b(wing)425 b(explicit)e(form)-36 b(ula)422 b(for)h(the)f(generalized)g (Bernoulli)h(p)36 b(olyno-)800 39304 y(mials)8519 42334 y SDict begin H.S end 8519 42334 a 8519 42334 a SDict begin 14.5 H.A end 8519 42334 a 8519 42334 a SDict begin [ /View [/XYZ H.V] /Dest (equation.22) cvn H.B /DEST pdfmark end 8519 42334 a -1574 w Fh(B)8000 41785 y Fi(\026)7933 42662 y(n)8621 42334 y Fk(\()p Fh(z)59 b Fk(\))443 b(=)12809 41435 y Fh(d)13485 40953 y Fi(n)p 12326 42028 2268 54 v 12326 43245 a Fh(dw)13968 42861 y Fi(n)15782 41435 y Fh(w)16748 40953 y Fi(\026)17369 41435 y Fh(e)17972 40953 y Fi(w)24 b(z)p 14859 42028 5265 54 v 14859 43245 a Fk(\()p Fh(e)15968 42861 y Fi(w)17018 43245 y Fe(\000)295 b Fk(1\))19502 42861 y Fi(\026)20478 42334 y Fe(j)20847 42533 y Fi(w)24 b Ff(=0)23305 42334 y Fk(=)369 b Fh(z)25349 41785 y Fi(n)26271 42334 y Fk(+)28309 40673 y Fi(n)27635 41072 y Fj(X)27578 43862 y Fi(m)p Ff(=1)29834 40461 y Fj(\022)30992 41435 y Fh(n)30811 43245 y(m)31949 40461 y Fj(\023)32927 42334 y Fh(z)33590 41785 y Fi(n)p Fd(\000)p Fi(m)36545 40673 y(m)36002 41072 y Fj(X)36144 43871 y Fi(j)51 b Ff(=0)38142 40461 y Fj(\022)39120 41435 y Fh(m)39382 43245 y(j)40258 40461 y Fj(\023)41235 42334 y Fk(\()p Fh(\026)p Fk(\))43030 42533 y Fi(m)p Fd(\000)p Fi(j)13494 47052 y Fe(\002)15492 45337 y Fi(j)14749 45790 y Fj(X)14850 48619 y Fi(k)24 b Ff(=0)16889 45179 y Fj(\032)17939 46153 y Fh(j)17885 47963 y(k)18606 45179 y Fj(\033)19602 47052 y Fk(\()p Fe(\000)p Fh(\026)p Fk(\))22430 47251 y Fi(k)23923 45392 y(k)23220 45790 y Fj(X)23432 48619 y Fi(l)11 b Ff(=0)25360 45179 y Fj(\022)26338 46153 y Fh(k)26490 47963 y(l)27059 45179 y Fj(\023\032)29033 46153 y Fk(2)p Fh(m)295 b Fe(\000)h Fh(j)29456 47963 y(m)f Fe(\000)g Fh(l)33057 45179 y Fj(\033)34610 46153 y Fk(\()p Fh(m)f Fe(\000)i Fh(l)29 b Fk(\)!)p 34186 46746 5398 54 v 34186 47963 a(\(2)p Fh(m)295 b Fe(\000)h Fh(j)75 b Fk(\)!)39717 47052 y Fh(:)p Black 10410 w Fk(\(22\))p Black 1302 50603 a(Comparing)502 b(it)g(with)f(the)h(form)-36 b(ula)502 b(giv)-36 b(en)502 b(b)-36 b(y)502 b(Sriv)-72 b(asta)-36 b(v)-72 b(a)502 b(and)f(T)-108 b(o)36 b(doro)-36 b(v)503 b([)p 0 .5 0 TeXcolorrgb 39384 50603 a SDict begin H.S end 39384 50603 a 0 .5 0 TeXcolorrgb Fk(11)p 0 .5 0 TeXcolorrgb 40685 49747 a SDict begin H.R end 40685 49747 a 40685 50603 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.ST) cvn H.B /ANN pdfmark end 40685 50603 a Black Fk(,)519 b(Eq.)502 b(\(3\),)519 b(p.)501 b(510],)521 b(w)-36 b(e)800 52209 y(see)434 b(that)f(form)-36 b(ula)434 b(\()p 0 .5 0 TeXcolorrgb 10996 52209 a SDict begin H.S end 10996 52209 a 0 .5 0 TeXcolorrgb Fk(22)p 0 .5 0 TeXcolorrgb 12296 51353 a SDict begin H.R end 12296 51353 a 12296 52209 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.22) cvn H.B /ANN pdfmark end 12296 52209 a Black Fk(\))g(do)36 b(es)434 b(not)f(in)-36 b(v)g(olv)g(e)435 b(an)-36 b(y)433 b(h)-36 b(yp)36 b(ergeometric)434 b(function.)2751 53814 y(In)f(particular,)h (for)g Fh(\026)369 b Fk(=)f(1)434 b(w)-36 b(e)434 b(obtain)g(the)f (common)g(Bernoulli)h(p)36 b(olynomials)7682 57102 y Fh(B)8670 57301 y Fi(n)9296 57102 y Fk(\()p Fh(z)59 b Fk(\))443 b(=)f Fh(z)13531 56554 y Fi(n)14453 57102 y Fe(\000)15914 56203 y Fk(1)p 15914 56796 651 54 v 15914 58013 a(2)16698 57102 y Fh(z)17361 56554 y Fi(n)p Fd(\000)p Ff(1)19485 57102 y Fk(+)21523 55442 y Fi(n)20849 55840 y Fj(X)20791 58630 y Fi(m)p Ff(=2)23047 55229 y Fj(\022)24206 56203 y Fh(n)24025 58013 y(m)25163 55229 y Fj(\023)26141 57102 y Fh(z)26804 56554 y Fi(n)p Fd(\000)p Fi(m)12868 61613 y Fe(\002)14122 59342 y Fj( )15174 61613 y Fh(m)16533 59953 y Fi(m)p Fd(\000)p Ff(1)16591 60352 y Fj(X)16693 63180 y Fi(k)24 b Ff(=1)18790 59740 y Fj(\032)19786 60715 y Fh(m)295 b Fe(\000)g Fk(1)21131 62525 y Fh(k)23197 59740 y Fj(\033)24194 61613 y Fk(\()p Fe(\000)p Fk(1\))26889 61065 y Fi(k)27458 61613 y Fh(k)45 b Fk(!)29464 59953 y Fi(k)28761 60352 y Fj(X)28974 63180 y Fi(l)11 b Ff(=0)30901 59740 y Fj(\022)31879 60715 y Fh(k)32032 62525 y(l)32600 59740 y Fj(\023\032)34574 60715 y Fh(m)295 b Fk(+)g(1)34681 62525 y Fh(m)f Fe(\000)i Fh(l)37964 59740 y Fj(\033)39200 60715 y Fk(\()p Fh(m)e Fe(\000)i Fh(l)29 b Fk(\)!)p 39094 61308 4763 54 v 39094 62525 a(\()p Fh(m)294 b Fk(+)h(1\)!)14302 66164 y(+)16078 64504 y Fi(m)15535 64902 y Fj(X)15636 67731 y Fi(k)24 b Ff(=1)17675 64291 y Fj(\032)18671 65266 y Fh(m)18880 67076 y(k)19809 64291 y Fj(\033)20805 66164 y Fk(\()p Fe(\000)p Fk(1\))23500 65616 y Fi(k)24069 66164 y Fh(k)45 b Fk(!)26076 64504 y Fi(k)25373 64902 y Fj(X)25585 67731 y Fi(l)11 b Ff(=0)27513 64291 y Fj(\022)28491 65266 y Fh(k)28643 67076 y(l)29212 64291 y Fj(\023\032)32206 65266 y Fh(m)31186 67076 y(m)295 b Fe(\000)g Fh(l)34364 64291 y Fj(\033)35493 65266 y Fk(\()p Fh(m)g Fe(\000)g Fh(l)29 b Fk(\)!)p 35493 65859 4551 54 v 37019 67076 a Fh(m)p Fk(!)40177 63893 y Fj(!)p Black 26475 70521 a Fk(8)p Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 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.9) cvn H.B /DEST pdfmark end 800 -4000 a Black Black 1234 -2672 a Fk(and)433 b(putting)f Fh(z)d Fk(=)368 b(0,)434 b(the)f(common)h(Bernoulli)g(n)-36 b(um)g(b)36 b(ers)12205 1434 y SDict begin H.S end 12205 1434 a 12205 1434 a SDict begin 14.5 H.A end 12205 1434 a 12205 1434 a SDict begin [ /View [/XYZ H.V] /Dest (equation.23) cvn H.B /DEST pdfmark end 12205 1434 a -1575 w Fh(B)11618 1633 y Fi(n)12687 1434 y Fk(=)443 b Fh(n)15212 -226 y Fi(n)p Fd(\000)p Ff(1)15139 172 y Fj(X)15241 3001 y Fi(k)24 b Ff(=1)17279 -439 y Fj(\032)18276 535 y Fh(n)295 b Fe(\000)h Fk(1)19440 2345 y Fh(k)21326 -439 y Fj(\033)22322 1434 y Fk(\()p Fe(\000)p Fk(1\))25017 885 y Fi(k)25586 1434 y Fh(k)45 b Fk(!)27593 -226 y Fi(k)26890 172 y Fj(X)27102 3001 y Fi(l)11 b Ff(=0)29030 -439 y Fj(\022)30008 535 y Fh(k)30160 2345 y(l)30729 -439 y Fj(\023\032)32703 535 y Fh(n)296 b Fk(+)e(1)32809 2345 y Fh(n)i Fe(\000)f Fh(l)35732 -439 y Fj(\033)36967 535 y Fk(\()p Fh(n)g Fe(\000)h Fh(l)29 b Fk(\)!)p 36861 1128 4402 54 v 36861 2345 a(\()p Fh(n)295 b Fk(+)g(1\)!)14142 5985 y(+)16048 4324 y Fi(n)15375 4723 y Fj(X)15476 7552 y Fi(k)24 b Ff(=1)17515 4112 y Fj(\032)18511 5086 y Fh(n)18539 6896 y(k)19287 4112 y Fj(\033)20284 5985 y Fk(\()p Fe(\000)p Fk(1\))22979 5436 y Fi(k)23548 5985 y Fh(k)45 b Fk(!)25554 4324 y Fi(k)24851 4723 y Fj(X)25064 7552 y Fi(l)11 b Ff(=0)26991 4112 y Fj(\022)27969 5086 y Fh(k)28121 6896 y(l)28690 4112 y 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