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On the Schröter formula for theta functions. (English) Zbl 1185.33026
Suppose that \(\tau\) is a complex number with positive imaginary part, and let \(q=e^{\pi i\tau}\). The Schröter formula in terms of the Jacobi theta function \(\theta_3(z|\tau)=1+2\sum_{n=1}^{\infty}q^{n^2}\cos(2nz)\) asserts that \[ \begin{split} \theta_3(rax+by|r\tau)\theta_3(-sbx+ay|s\tau)\\ =\sum_{k=0}^{m-1}q^{rk^2}e^{2ki(rax+by)}\theta_3(mx+rak\pi\tau|m\tau)\theta_3(my+rsbk\pi\tau|rsm\tau), \end{split} \] where \(a,b,r\) and \(s\) are positive integers, \(m=ra^2+sb^2\), and \(\gcd(b,m)=1\). In this paper, the authors study various identities of theta functions using Schröter formula. More precisely, they prove some identities concerning products of two theta functions and their sums, which two of them reprove two well-known Ramanujan identities related to the modular equation of degree 5. Also, the Hirschhorn septuple product identity is obtained as a special case of Schröter formula.

33E05 Elliptic functions and integrals
11F27 Theta series; Weil representation; theta correspondences
11P84 Partition identities; identities of Rogers-Ramanujan type
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