Applications of Lipschitz summation formula and a generalization of Raabe's cosine transform

TL;DR

This paper develops a new transformation formula based on the Lipschitz summation formula, generalizes Ramanujan's and Wright's results, and introduces a novel generalization of Raabe's cosine transform with applications in analysis and number theory.

Contribution

It introduces a new transformation formula generalizing Ramanujan's and Wright's results, and a novel generalization of Raabe's cosine transform, with applications in analysis and number theory.

Findings

Derived a new transformation formula generalizing Ramanujan's result.

Provided a simpler proof for a non-modular transformation related to divisor sums.

Introduced a generalized Raabe's cosine transform with several properties demonstrated.

Abstract

General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan's result, in turn, is a generalization of the modular transformation of Eisenstein series $E_k(z)$ on SL$_2(\mathbb{Z})$, where $z\to-1/z, z\in\mathbb{H}$. The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with Kesarwani giving a non-modular transformation for $\sum_{n=1}^{\infty}\sigma_{2m}(n)e^{-ny}$ is also derived using the Lipschitz summation formula. In the pursuit of obtaining this transformation, we naturally encounter a new generalization of Raabe's cosine transform whose several properties are also demonstrated. As a corollary of this result, we get a generalization of Wright's asymptotic estimate for the generating function of the number of plane partitions of a positive integer $n$.