MAPLE PROGRAMS 

§1 Proposition 3

S := proc(m,n) sum((-1)^i*binomial(m,i)*binomial(n,i), i=0.. n) end;
IP0 := proc(n) irreduc(simplify(expand(S(m,2*n)))) end;

for n from 0 to 300 do print(n,IP0(n)) od;
0, false
memory used=11695.3MB, alloc=242.1MB, time=102.57
                                   100, true
memory used=27609.8MB, alloc=252.1MB, time=238.27
                                   120, true
memory used=3765820.6MB, alloc=1571.5MB, time=35101.02
                                   300, true
                                   
IP1 := proc(n) irreduc(simplify(expand(S(m,2*n+1))/(m-2*n-1))) end;
for n from 0 to 119 do print(n,IP1(n)) od;
0, false
memory used=1278345.9MB, alloc=297.4MB, time=15841.70
                                   99, true
memory used=5585485.5MB, alloc=496.5MB, time=63908.41
                                   119, true

§2.1

Inequality (8)
simplify(diff(  (r-1-(r^2-6*r+1)^(1/2))/(2*r) , r));

Inequality (12)
M :=  (1-2*x-x^2)/(1+x)/(1-x)^2;
simplify(diff(M,x)*(1-x)^3*(1+x)^2);

§2.2

To get estimate (18)
ff1 := cos(x)-1+x^2/2-(1-cos(x))^2/6-2*(1-cos(x))^3/45 ;
gg1:= simplify(diff(diff(ff1,x),x));

End of proof of Lemma 12
simplify((1-2*x-x^2)/(1+x)/(1-x)^2/6-x^2/(1-x)^4-x/(1-x)/(1+x)^3);
c1 := (1-8*x+9*x^2-32*x^3-9*x^4-8*x^5-x^6)/6/(1+x)^3/(1-x)^4;
c2 := (1-2*x-x^2)/(1+x)/(1-x)^2*(2/45+Pi^2/45)+4/3*(x^3/(1-x)^6-x^2/(1+x)^5/(1-x));
fsolve(c1+c2/2=0); 0.191862212579945
subs(x=0,simplify(diff(c1+c2/2,x)));
fsolve(simplify(diff(c1+c2/2,x)));

End of proof of Lemma 13
factor(-(1-2*x-x^2)/3/(1+x)/(1-x)^2+x^2/(1-x)^4+x/(1-x^2)/(1+x)^2);
num := 3*(x-1)^4*(1+x)^3*factor(-(1-2*x-x^2)/3/(1+x)/(1-x)^2+x^2/(1-x)^4+x/(1-x^2)/(1+x)^2);
fsolve(num=0);0.20027332371160260475
fsolve(diff(num, x)=0);-4.111612121

End of proof of Lemma 14
ff3 := (1+x^2)*(x^2+4*x-1)/(1-x)^3/(1+x)^2;
ff4 := 1/(1+x^2+2*c*x)-x*(1+x)/(1-x)/(1+x^2-2*x*c);
factor(ff3+ff4);
factor( ff3-subs(c=1/2,ff4));
num := (x-1)^3*(1+x)^2*(x^2+x+1)*(x^2-x+1)*factor( ff3-subs(c=1/2,ff4))/(1+x^2);
fsolve(num=0);0.262108902129679
fsolve(diff(num, x)=0);-3.172308262

§2.3

ff2 := (x^4+6*x^3+2*x^2+4*x-1)/6/(1+x)/(1-x)^4+x/(1-x^2)/(1-x)^2;
factor(diff(ff2,x));
evala(subs(x=2^(1/2)-1,ff2));

ff3 := (1+x^2)*(x^2+4*x-1)/(1-x)^3/(1+x)^2;
ff4 := 1/(1+x^2+2*c*x)-x*(1+x)/(1-x)/(1+x^2-2*x*c);
factor(ff3-subs(c=-1,ff4));
simplify(diff(subs(c=-1,ff4),x));
evala(subs(x=2^(1/2)-1,c=-1,ff4));

§2.4

First inequality
R := proc(r) (r-1-(r^2-6*r+1)^(1/2))/(2*r) end;
M := proc(r) evalf((1-2*R(r)-R(r)^2)/(1+R(r))/(1-R(r))^2) end;
F1 := proc(r,x) evalf(3*(3+2^(3/2))*Pi^5/256/x/M(r)^2+5*(3+2^(3/2))/24/x/M(r)^3+2^(1/2)*exp(-x*M(r)*Pi^2/2)*Pi^(-3/2)*(x*M(r))^(-1/2)) end;
F1(5.941893,241)=0.99999785017193857323

Monotonicity 
Cg := (x^4+6*x^3+2*x^2+4*x-1)/24/(1+x)/(1-x)^4+x/4/(1-x^2)/(1+x^2+2*x*c);
factor(diff(Cg,x));
Ch := (1+x^2)*(x^2+4*x-1)/6/(1-x)^3/(1+x)^2;
factor(diff(Ch,x));

End of the section
R := proc(r) (r-1-(r^2-6*r+1)^(1/2))/(2*r) end;
M := proc(r) (1-2*R(r)-R(r)^2)/(1+R(r))/(1-R(r))^2 end;
Cg := proc(x,c) (x^4+6*x^3+2*x^2+4*x-1)/24/(1+x)/(1-x)^4+x/4/(1-x^2)/(1+x^2+2*x*c) end;
Ch := proc(x) (1+x^2)*(x^2+4*x-1)/6/(1-x)^3/(1+x)^2 end;
F2 := proc(r,x,a) evalf( 3*Cg(R(r),cos(a))/x/M(r)^2+15*Ch(R(r))^2/2/x/M(r)^3+((2/Pi/x/M(r))^(1/2)/a+(1-a/Pi)*(2*Pi*x*M(r))^(1/2))*exp(-x*M(r)*a^2/2)) end;
Cond := proc(r,x,a) evalf(x*M(r)*a^2) end;

F2(5.8478,241,0.75);0.9935952839
Cond(5.8478,241,0.75);20.40053456
F2(5.8362,980,0.5);0.9998405530
Cond(5.8362,980,0.5);24.17370668

S := proc(m,n) sum((-1)^i*binomial(m,i)*binomial(n,i), i=0.. n) end;
T := proc(n) for m from ceil(5.8362*n) to floor(5.8478*n) do if S(m,n)=0 then print(m,n) fi od end;
for n from 241 to 979 do T(n) od;
memory used=682640.7MB, alloc=106.3MB, time=5470.53

§3.1
Formula (21)
evala(subs(z=(r-1+I*(-r^2+6*r-1)^(1/2))/2/r,(1-2*z-z^2)/(1+z)/(1-z)^2));
factor(r^3-5*r^2-5*r+1);
simplify((r+1)^2*(-r^2+6*r-1)/16/r^2+(r-1)^4/16/r^2);

§3.2
End of Lemma 16
A := ((-r^2+6*r-1)^(1/2)+r^(1/2)*(r-1))/2/(r^2+1+r^(1/2)*(r-1))^(1/2)+(-r^2+6*r-1)^(1/2)/4;
B := r^2/(r+1)^2*A ;
for i from 100 to 582 do print(i,evalf(subs(r=i/100,B))) od;
fsolve(diff(B,r)=0);4.46452368746048
Bmax= subs(r=4.46452368746048,B);1.01536714165069

End of Lemma 17
ff5 := (1-c^2)*(x-a)^2-(cos(x)-c)^2-4*c/3*(cos(x)-c)^3;
gg5:= simplify(diff(diff(ff5,x),x));
p := 6*x^2*y+2*(1-y^2)*x-3*y;
minimize(p,x=-1..1,y=0..1, location);

§3.3
A := 24.45*exp(2)*(-r^2+6*r-1)+16*r^2*(-r^2+6*r-1)*Pi^(-1/2)/(r+1)+864*Pi^(1/2)*r^3*exp(-1)/(r+1);
for i from 101 to 582 do print(i,evalf(subs(r=i/100,A))) od;
fsolve(diff(A,r));33.42645125
evalf(subs(r=3+2^(3/2),A));16335.31939

§4.1

Cpp := proc(l) simplify(expand(   sum((-1)^j*binomial(2*l,2*j)*pochhammer(k+1+j,l-j)*pochhammer(k+1-j,j),j=0..l  )    )) end;
Cpp(0)= 1                                     
Cpp(1)= 1
2,2,4,4
Cpp := proc(l) irreduc(simplify(expand(   sum((-1)^j*binomial(2*l,2*j)*pochhammer(k+1+j,l-j)*pochhammer(k+1-j,j),j=0..l  )    ))) end;
for l from 0 to 500 do print(l,Cpp(l)) od;
 0, false      1, false
memory used=10474414.0MB, alloc=4965.7MB, time=95201.16
                                   350, true

Cpi := proc(l) simplify(expand(   sum((-1)^j*binomial(2*l+1,2*j)*pochhammer(k+1+j,l-j)*pochhammer(k+1-j,j),j=0..l  )    )) end;
Cpi(0)= 1
Cpi(1)=-2k+1
2,3,4,5
Cpi := proc(l)  irreduc(simplify(expand(   sum((-1)^j*binomial(2*l+1,2*j)*pochhammer(k+1+j,l-j)*pochhammer(k+1-j,j),j=0..l  )    ))) end;
for l from 0 to 500 do print(l,Cpi(l)) od;
0, false
memory used=10466095.3MB, alloc=4997.7MB, time=98276.16
                                   350, true

Cip := proc(l) simplify(expand(  sum((-1)^j*binomial(2*l,2*j+1)*pochhammer(k+2+j,l-1-j)*pochhammer(k+1-j,j),j=0..l-1  )     )) end;
Cip(0)=0
Cip(1)=2
Cip(2)=8
2,2,4,4,6,6
Cip := proc(l)  irreduc(simplify(expand(  sum((-1)^j*binomial(2*l,2*j+1)*pochhammer(k+2+j,l-1-j)*pochhammer(k+1-j,j),j=0..l-1  )     ))) end;
for l from 0 to 500 do print(l,Cip(l)) od;
0, false    1, false    2, false
memory used=4721176.1MB, alloc=2981.2MB, time=42161.91
                                   300, true
for l from 301 to 400 do print(l,Cip(l)) od;  
memory used=5559544.0MB, alloc=2405.7MB, time=42970.55
                                   350, true                          
                                  
Cii := proc(l) simplify(expand(   sum((-1)^j*binomial(2*l-1,2*j+1)*pochhammer(k+2+j,l-1-j)*pochhammer(k+1-j,j),j=0..l-1  )     )) end;
Cii(0)=0
Cii(1)=1
Cii(2)=2k+6
2,3,4,5
Cii := proc(l)  irreduc(simplify(expand(   sum((-1)^j*binomial(2*l-1,2*j+1)*pochhammer(k+2+j,l-1-j)*pochhammer(k+1-j,j),j=0..l-1  )     ))) end;
for l from 0 to 500 do print(l,Cii(l)) od;
0, false                       1, false
memory used=10457825.1MB, alloc=5029.7MB, time=97200.48
                                   351, true

§4.3
Beginning of proof of Lemma 19
f := arccos((r-1)/2/r^(1/2))-(r-1)^(1/2);
simplify(diff(f,r));
fsolve(f=0);2.282459833

Middle of proof of Lemma 19
fsolve( -r^2+6*r-1-(r-3)^2*(r-1));4.355301398

Beginning of proof of Lemma 20
f := (r^2-4*r-1)/16/r+arccos((r-1)/2/r^(1/2))/8;
simplify(diff(f,r));
fsolve( (r^2+r)^2-(r^2+1)^2*(-r^2+6*r-1));0.1789720897, 5.587463394
fsolve(f=0);2.119518709
for i from 100 to 300 do print(i, evalf(subs(r=i/100,f))) od;

§4.4

a := proc(r) arccos((r-1)/2/r^(1/2)) end;
FF4 := proc(r,x) evalf( 1/3/(2*Pi*x)^(1/2)/((3-r)/4-(r-1)^(1/2)/3)^2+(r-1)/2^(7/2)/((3-r)/4-(r-1)^(1/2)/3)^(3/2)+(2/Pi/x/(r-1))^(1/2)*exp(-x*(r-1)/2)+(0.6267+0.04206*x^(1/2))*x^(1/2)*exp(-x*(1-Pi/4)*a(r)^2/8)) end;
ff41 := proc(r,x) evalf( 1/3/(2*Pi*x)^(1/2)/((3-r)/4-(r-1)^(1/2)/3)^2+(r-1)/2^(7/2)/((3-r)/4-(r-1)^(1/2)/3)^(3/2)+(2/Pi/702)^(1/2)*exp(-351)+(0.6267+0.04206*x^(1/2))*x^(1/2)*exp(-x*(1-Pi/4)*a(r)^2/8)) end;
ff41(1.03581,19609);0.01649011716
ff42 := proc(r,x,c) evalf( 1/3/(2*Pi)^(1/2)/((3-r)/4-(r-1)^(1/2)/3)^2+c/2^(7/2)/((3-r)/4-(r-1)^(1/2)/3)^(3/2)+(1/351/Pi)^(1/2)+(0.6267+0.04206*x^(1/2))*x*exp(-x*(1-Pi/4)*a(r)^2/8)) end;
gg42 := proc(c) ff42(1+c*ceil(702^2/c^2)^(-1/2),ceil(702^2/c^2),c) end;
gg42(1.67007);1.05881471960742
ff43 := proc(r,x,c) evalf( 1/3/(2*Pi)^(1/2)/((3-r)/4-(r-1)^(1/2)/3)^2+Pi^(1/2)/2^(7/2)/((3-r)/4-(r-1)^(1/2)/3)^(3/2)+(2/Pi/c/x^(1/2))^(1/2)*exp(-c*x^(1/2)/2)+(0.6267+0.04206*x^(1/2))*x*exp(-x*(1-Pi/4)*a(r)^2/8)) end;
gg43 := proc(c) ff43(1+evalf(c*156865^(-1/2)),156865,c) end;
gg43(1.67007);1.05881750978616
ff44 := proc(r,x,k) evalf( 1/3/(2*Pi)^(1/2)/((3-r)/4-(r-1)^(1/2)/3)^2+((k+1)*Pi)^(1/2)/2^(7/2)/((3-r)/4-(r-1)^(1/2)/3)^(3/2)+(2/Pi/(k*Pi*x)^(1/2))^(1/2)*exp(-(k*Pi*x)^(1/2)/2)+(0.6267+0.04206*x^(1/2))*x*exp(-x*(1-Pi/4)*a(r)^2/8)) end;
for k to 7 do print(k,ff44(1+evalf(((k+1)*Pi/ceil(702^2/(k+1)/Pi))^(1/2)),evalf(ceil(702^2/(k+1)/Pi)),k)) od;
 1, 1.307745617
 2, 1.509288944
 3, 1.688759306
 4, 1.854819247
 5, 2.011886003
 6, 2.162592902
 7, 2.308644312

§4.5
Proof of Lemma 21
gamma1 := arccos((3*x-1)/2^(3/2)/x);
factor(diff(gamma1,x));
gamma2:=-arccos((x-3)/2^(3/2));
factor(diff(gamma2,x));
G3 := simplify(diff(gamma1,x$2));
G4 := simplify(diff(gamma1,x$3));
fsolve(3*x^4-27*x^3+70*x^2-15*x+1=0);

§4.6
§4.6.1
v01 := proc(c,x) evalf(Pi/2-1.30775*(x-1.30775^2)^(-1/2)-((Pi/2)^(1/2)-c*x^(-1/2))^2) end;
v010 := proc(c) v01(c,ceil((702+2*c)^2/2/Pi)) end;
v010(0.52215); >0
v010(0.5221); <0

v02 := proc(c,x) evalf( Pi/2-Pi/2*(1-(3*Pi/4/x)^(1/2))*(1+c*(2*Pi*x)^(-1/2))^2+1.50929*(x-1.50929^2)^(-1/2)) end;
v02(3.14063,52289); <0
v02(3.14062,52289); >0

v03 := proc(c,x) evalf(3*Pi/2*((1-c*(6*Pi*x)^(-1/2))^2-1)+ 2.01189*(x- 2.01189^2)^(-1/2)) end;
v030 := proc(c) v03(c,ceil((702+c)^2/6/Pi)) end;
v030(0.9275); <0
v030(0.9274); >0

§4.6.2
v11 := proc(c,x) evalf(3*Pi/4-1.50929*(x-1.50929^2)^(-1/2)-((3*Pi/4)^(1/2)-c*x^(-1/2))^2) end;
v110 := proc(c) v11(c,ceil((703+2*c)^2/3/Pi)) end;
v110(0.492); >0
v110(0.4919); <0

v12  := proc(c,x) evalf( 3*Pi/4-3*Pi/4*(1-(Pi/x)^(1/2))*(1+c*(3*Pi*x)^(-1/2))^2+1.68876*(x-1.68876^2)^(-1/2)) end;
v12(3.8433,39217); <0
v12(3.8432,39217); >0

v13 := proc(c,x) evalf(7*Pi/4*(1-c*(7*Pi*x)^(-1/2))^2-7*Pi/4+2.1626*(x-2.1626^2)^(-1/2)) end;
v130 := proc(c) v13(c,ceil((703+c)^2/7/Pi)) end;
v130(0.9231); <0
v130(0.923); >0

§4.6.3
v21 := proc(c,x) evalf( -(1-c/2*x^(-3/4))*c^2/4*x^(-1/2)+1.05882*(x-1.05882^2)^(-1/2)) end;
v21(2.0582,156865); <0
v21(2.0581,156865); >0

v22 := proc(c,x) evalf(Pi*(1-c/2*(Pi*x)^(-1/2))^2-Pi+1.68876*(x-1.68876^2)^(-1/2)) end;
v220  := proc(c) v22(c,ceil((702+c)^2/4/Pi)) end;
v220(0.9535); <0
v220(0.9534); >0

v23  := proc(c,x) evalf( Pi-Pi*(1-(5*Pi/4/x)^(1/2))*(1+c*(4*Pi*x)^(-1/2))^2+1.85482*(x-1.85482^2)^(-1/2)) end;
v23(4.5938,31373);

v24 := proc(c,x) evalf(2*Pi*(1-c/2*(2*Pi*x)^(-1/2))^2-2*Pi+2.30865*(x-2.30865^2)^(-1/2)) end;
v240  := proc(c) v24(c,ceil((702+c)^2/8/Pi)) end;
v240(0.9218); <0
v240(0.9217); >0

§4.6.4
v30 := proc(c) evalf( Pi/4-1.05882*(ceil((703+2*c)^2/Pi)-1.05882^2)^(-1/2)-((Pi/4)^(1/2)-c*ceil((703+2*c)^2/Pi)^(-1/2))^2) end;
v30(0.5979); >0
v30(0.5978); <0

v32  := proc(c,x) evalf( Pi/4-Pi/4*(1-(Pi/2/x)^(1/2))*(1+c*(Pi*x)^(-1/2))^2+1.30775*(x-1.30775^2)^(-1/2)) end;
v32(2.5913,78656); <0
v32(2.5912,78656); >0

v33 := proc(c,x) evalf(5*Pi/4*(1-c*(5*Pi*x)^(-1/2))^2-5*Pi/4+1.85482*(x-1.85482^2)^(-1/2)) end;
v330 := proc(c) v33(c,ceil((703+c)^2/5/Pi)) end;
v330(0.9367); <0
v330(0.9366); >0