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

2022/09/26 by Atul Dixit, Rahul Kumar, Dixit, Atul +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.12658

openalex publication_date 2022/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 SL₂(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^∞σ_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.

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