On an Identity Associated With Jacobi’s Theta Functions

Recently, Yang [‘On modular equations for the Jacobi theta functions’, Preprint] discovered an identity that involves the sum of squares of the theta functions ϑ j ( 0 | τ ) ϑ j ( 0 | N τ ) ϑ ⱼ(0|τ )ϑ ⱼ(0|Nτ ) theta Subscript j Baseline left parenthesis 0 vertical bar tau right parenthesis theta Subscript j Baseline left parenthesis 0 vertical bar upper N tau right parenthesis for j = 2 , 3 , 4. $j=2,3,4.$ j equals 2 comma 3 comma 4 period In this article, we prove this identity and its generalisations. Our study also leads us to a four-variable extension of an identity discovered by Chan, Chua and Solé [‘Quadratic iterations to π π pi associated to elliptic functions to the cubic and septic base’, Trans. Amer. Math. Soc. 355 (4) (2002), 1505–1520].