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GENERALIZED LAMBERT SERIES, RAABE’S COSINE TRANSFORM AND A GENERALIZATION OF RAMANUJAN’S FORMULA FOR $\unicode[STIX]{x1D701}(2m+1)$

  • ATUL DIXIT (a1), RAJAT GUPTA (a2), RAHUL KUMAR (a3) and BIBEKANANDA MAJI (a4)
Abstract

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$ , $N\in \mathbb{N}$ and $h\in \mathbb{Z}$ , is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for $\unicode[STIX]{x1D701}(2m+1)$ for $m>0$ , the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for $\unicode[STIX]{x1D701}(1/N),N$ even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series $E_{2}(z)$ and $E_{2}(-1/z)$ . An identity relating $\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$ . In particular, this gives a beautiful relation between $\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$ and $\unicode[STIX]{x1D701}(11)$ . New results involving infinite series of hyperbolic functions with $n^{2}$ in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.

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