Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for $\hat{I}\P (2m+1)$

Abstract

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\exp (-a{n}^{N}x)}{1-\exp (-n^{N}x)},0<a\le 1,x>0$, $N\in \mathbb{N}$ and $h\in \mathbb{Z}$, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan's famous formula for $\hat{I}\P (2m+1)$, $m>0$ and the transformation formula for $\log \hat{I}\cdot (z)$. Numerous important special cases of our transformations are derived. An identity relating $\hat{I}\P (2N+1),\hat{I}\P (4N+1),\cdots ,\hat{I}\P (2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. Certain transcendence results of Zudilin- and Rivoal-type are obtained for odd zeta values and generalized Lambert series. A criterion for transcendence of $\hat{I}\P (2m+1)$ and a Zudilin-type result on irrationality of Euler's constant ${\hat{I}}^3$ are also given. New results analogous to those of Ramanujan and Klusch for $N$ even, and a transcendence result involving $\hat{I}\P \left(2m+1-\frac{1}{N}\right)$, are obtained.

48 pages, submitted for publication, comments are welcome