Treffer: AN ANALOGUE OF AN IDENTITY OF JACOBI

H. H. Chan, K. S. Chua and P. Solé [‘Quadratic iterations to $\pi $ associated to elliptic functions to the cubic and septic base’, Trans. Amer. Math. Soc.355(4) (2002), 1505–1520] found that, for each positive integer d, there are theta series $A_d, B_d$ and $C_d$ of weight one that satisfy the Pythagoras-like relationship $A_d^2=B_d^2+C_d^2$ . In this article, we show that there are two collections of theta series $A_{b,d}, B_{b,d}$ and $C_{b,d}$ of weight one that satisfy $A_{b,d}^2=B_{b,d}^2+C_{b,d}^2,$ where b and d are certain integers.