Some identities involving theta functions
Let \(q\) be a complex variable with \(|q|<1\). Let \(\varphi(q):=\sum_{n=-\infty}^{\infty}q^{n^2}\) be the theta function, and for \(k\geq 1\) let \[ F_k(q):=1+\frac{4}{E_{2k}}\sum_{n=1}^\infty\left(\sum_{d|n}\left(\frac{-4}{d}\right)d^{2k}\right)q^n, \] where \(E_{2k}\) is the Euler number and \(\left(\frac{-4}{d}\right)\) is the Legendre-Jacobi-Kronecker symbol, \[ G_k(q):=\sum_{n=1}^\infty \left(\sum_{d|n}\left(\frac{-4}{n/d}\right)d^{2k}\right)q^n. \] The authors prove that \(F_k(q)\) and \(G_k(q)\) can be expressed as a polynomial in \(\varphi(q)\) and \(\varphi(-q)\) with rational coefficients and give recurrence relations for these coefficients.
scientific article; zbMATH DE number 5559088
scientific article