Jacobi Triple Product

Author

Eric W. Weisstein
September 21, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/JacobiTripleProduct.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/JacobiTripleProduct.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Identity

(∑_(m=-∞)^∞)(x^(m^2)) (z^(2 m))

Sum[(x^(m^2))(z^(2m)),{m,-∞,∞}]

(∑_(k=-∞)^∞)(q^(k (k+1)/2)) (x^k)

Sum[(x^k)(q^(k(k+1)/2)),{k,-∞,∞}]

(∑_(n=-∞)^∞)(((-1))^n) (a^n) (q^((1/2) n (n-1)))

Sum[(((-1))^n)(a^n)(q^(n(n-1)/2)),{n,-∞,∞}]

(∑_(n=0)^∞)(((-1))^n) (2 n+1) (q^(n (n+1)/2))==((QPochhammer1[q])^3)

Sum[(((-1))^n)(2n+1)(q^(n(n+1)/2)),{n,0,∞}]

Sum[(((-1))^n)(2n+1)(q^(n(n+1)/2)),{n,0,10}]

Series[((QPochhammer[q])^3),{q,0,55}]//Normal

(1/2) (∑_(n=-∞)^∞)(((-1))^n) (2 n+1) (q^(n (n+1)/2))==(QPochhammer1[q])

TimeConstrained[(1/2)Sum[(((-1))^n)(2n+1)(q^(n(n+1)/2)),{n,-∞,∞}],60]//Timing

(1/2)Sum[(((-1))^n)(2n+1)(q^(n(n+1)/2)),{n,-10,10}]//Simplify

Series[((QPochhammer[q])^3),{q,0,55}]//Normal

(∏_(n=1)^∞)((((x^(2 n-1))+1))^2) (1-(x^(2 n)))==(Subscript[ϑ, 3])(0,x)

V?

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

FullSimplify[%]

FunctionExpand[%]//FullSimplify

{(Sqrt[(2/π)]) (Sqrt[EllipticK[InverseEllipticNomeQ[x]]]) ((InverseEllipticNomeQ[x])^(1/4)),EllipticTheta[3,0,x]}/.x->.4

FullSimplify[(Sqrt[(2/π)]) (Sqrt[EllipticK[InverseEllipticNomeQ[x]]]) ((InverseEllipticNomeQ[x])^(1/4))-EllipticTheta[3,0,x]]

FullSimplify[(Sqrt[(2/π)]) (Sqrt[EllipticK[(((EllipticTheta[2,0,x])^4)/((EllipticTheta[3,0,x])^4))]]) ((((((EllipticTheta[2,0,x])^4)/((EllipticTheta[3,0,x])^4))))^(1/4))-EllipticTheta[3,0,x]]

V3.0

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

V4.2

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

V5.1

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

V6

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

V7

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

V9

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

V10

$Version

Product[(((1+(x^(2n-1))))^2)(1-(x^(2n))),{n,1,∞}]

(∏_(n=1)^∞)((((x^(2 n-1))+1))^2)==(((x^2) ((((QPochhammer2[-(1/x), (x^2)])))^2))/(((x+1))^2))

V10

$Version

Product[(((1+(x^(2n-1))))^2),{n,1,∞}]