Applications of the Lipschitz Summation Formula and a Generalization of Raabe’s Cosine Transform

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 Ek(z) on SL 2(Z) , where z→ - 1 / z, z∈ 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 ∑n=1∞σ2m(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 an application of our results, we get a generalization of Wright’s asymptotic estimate for the generating function of the number of plane partitions of a positive integer n. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.