\r hypergeom.inc.gp

default(echo,1);
default(realprecision,55);
default(format,"g.25");


N=1130;

\\k=1/sqrt(2)    \\ k_1
k=3-2*sqrt(2)  \\ k_4
kp=sqrt(1-k^2)
K=Pi/2*hypergeom([1/2,1/2],[1],k^2,N)
Kp=Pi/2*hypergeom([1/2,1/2],[1],kp^2,N)

rat=Kp/K
q=exp(-Pi*rat)

\\ eta-minus:
em=prod(j=1,N+1,1-q^j)
t1=em^6
t2=4/Pi^3*k^(1/2)*kp^2*K^3  *q^(-1/4)
t1-t2  \\ == zero
\\
t1=em^24
t2=256/Pi^12*k^(2)*kp^8*K^12  *q^(-1)
t1-t2  \\ == zero


\\ for k_4 only:
t1=em*q^(1/24)
t2=(gamma(1/4)/(2*Pi^(3/4)))
t1-t2  \\ == zero

\\ http://mathworld.wolfram.com/EllipticIntegralSingularValue.html
K-((sqrt(2)+1)*gamma(1/4)^2)/(2^(7/2)*sqrt(Pi))


t1=em*q^(1/24)
t2=(256/Pi^12*k^(2)*kp^8*K^12)^(1/24)
t2=(256/Pi^12*k^(2)*kp^8*( ((sqrt(2)+1)*gamma(1/4)^2)/(2^(7/2)*sqrt(Pi)) )^12)^(1/24)
t2=gamma(1/4)*(256/Pi^12*k^(2)*kp^8*( ((sqrt(2)+1))/(2^(7/2)*sqrt(Pi)) )^12)^(1/24)
t2=gamma(1/4)/Pi^(3/4)*(256*k^(2)*kp^8*( ((sqrt(2)+1))/(2^(7/2)) )^12)^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 256^(1/24) *(k^(2)*kp^8* ( ((sqrt(2)+1))/(2^(7/2)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(1/3) *(k^(2)*kp^8* ( ((sqrt(2)+1))/(2^(7/2)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(1/3)*(1/(2^(7/2)))^(1/2) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(1/3)*(1/(2^(7/4))) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(1/3)*(2^(-7/4)) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(-17/12) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(-1-5/12) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t2=gamma(1/4)/Pi^(3/4)* 2^(-1)*  2^(-5/12) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t1-t2  \\ == zero

t3= 2^(-5/12) *(k^(2)*kp^8* ( ((sqrt(2)+1)) )^12 )^(1/24)
t3= 2^(-5/12) *(k^(2)*kp^8)^(1/24) * (sqrt(2)+1)^(1/2)
t3= 2^(-5/12) *k^(1/12)*kp^(1/3) * (sqrt(2)+1)^(1/2)

quit; /* +++++++++++++++++++++++ */


\\ eta-plus:
ep=prod(j=1,N+1,1+q^j)
t1=ep^6
t2=1/2 * k^(1/2) * kp^(-1)  *q^(-1/4)
t1-t2  \\ == zero
\\
t1=ep^24
t2=1/16 * k^(2) * kp^(-4)  *q^(-1)
t1-t2  \\ == zero


\\ prelim in Whittaker/Watson p.488:
t1=(k/kp)^(1/2)
t2=2*q^(1/4)* ( prod(n=1,N+1,1+q^(2*n)) / prod(n=1,N+1, 1-q^(2*n-1)) )^2
t1-t2  \\ == zero

t1=2/Pi * K * k^(1/2)
t2=2*q^(1/4)* ( prod(n=1,N, 1-q^(2*n)) / prod(n=1,N, 1-q^(2*n-1)))^2
t1-t2  \\ == zero


quit; /* +++++++++++++++++++++++ */