More generating functions for values of certain \(L\)-functions (Q2905218)

The author proves some identities on generating functions for values of \(L\)-functions related to work of \textit{G. E. Andrews} et al. [Duke Math. J. 108, No. 3, 395--419 (2001; Zbl 1005.11048)] and \textit{D. Corson} et al. [J. Number Theory 107, No. 2, 392--405 (2004; Zbl 1056.11056)]. Sample result:NEWLINENEWLINETheorem 1.5. As a formal power series in \(t\), we have NEWLINE\[NEWLINEe^{-t/8} \sum_{n=0}^\infty (-1)^n \frac{(1-e^{-2t})(1-e^{-4t})\dots (1-e^{-(2n)t})}{(1+e^{-t})(1+e^{-3t})\dots(1+e^{-(2n+1)t})}=\frac 12 \sum_{n=0}^\infty \left(-\frac 18\right)^n \lambda_K(-n)\frac{t^n}{n!},NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\lambda_K(s)=\sum_{_{\substack{ a\subseteq O_K\\ N(a)\equiv 1\pmod 8}}} \chi(a)N(a)^{-s},NEWLINE\]NEWLINE denotes the Hecke \(L\)-function for \(K=\mathbb Q(\sqrt 2)\), and \(\chi(a)=(-1)^{(N(a)-1/8}\) when \(N(a)\equiv 1\pmod 8\).