Note on trigonometric expansions of theta functions
Let \(\theta_4(\nu,\tau) = \sum_{n=-\infty}^\infty (-1)^nq^{n^2}e^{2\pi in\nu}\), where \(q=e^{i\pi\tau}\). Then it is well known that \[ \log\frac{\theta_4(\nu,\tau)}{\theta_4(0,\tau)} = 2\sum_{n=1}^\infty \frac{q^n}{n(1-q^{2n})}(1-\cos 2n\pi\nu). \] One of the main results of this article is an expansion in powers of \(\sin\pi\nu\), namely \[ \log\frac{\theta_4(\nu,\tau)}{\theta_4(0,\tau)} =\sum_{p=1}^\infty c_{2p}(\tau)(\sin\pi\nu)^{2p}. \] Explicit formulas for the coefficients \(c_{2p}(\tau)\) are obtained by expanding \(\cos 2n\pi\nu\) as a polynomial in \(\sin \pi \nu\). The other main result is a system of differential recurrence relations for the coefficients \(c_{2p}(\tau)\). These relations are obtained from the partial differential equation (i.e., the heat equation) satisfied by \(\theta_4(\nu,\tau).\)
scientific article; zbMATH DE number 1915226