Some identities of Ramanujan type
For every pair of integers \(u\), \(v\), \(9\leq u<v\), we define functions NEWLINE\[NEWLINEg_{u,v}(\tau)= \sum^\infty_{n=1} n^u\sigma_{-v}(n) q^n,NEWLINE\]NEWLINE where \(\tau\in\mathbb C\), \(\text{Im\,}\tau> 0\), and \(q=e^{2\pi ir}\), \(\sigma_m(n)= \sum_{d|n} d^m\). The author proves the following identity: NEWLINE\[NEWLINEg_{u,v}(-1/\tau)= (-1)^{\frac{v-1}{2}-u} \sum^u_{l=0} {u!\over l!} {v-l-1\choose v-u-1} (-i\tau)^{1-v+ u+l}(2\pi)^{- u+l} g_{l,v}(\tau) + R(\tau,u,v),NEWLINE\]NEWLINE where \(v\geq 3\) is odd, and we omit the explicit expression for \(R(\tau,u,v)\) (due to its length). This result generalizes some classical identities of Ramanujan and \textit{E. Grosswald} [Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl. 1970, 9--13 (1970; Zbl 0206.05902)].
scientific article; zbMATH DE number 6077900