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

Abstract

General summation formulas have been proved to be very useful in 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 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 transforming ∑∞n=1σ2m(n)e-ny is also derived using the Lipschitz summation formula. In this pursuit, we naturally encounter a new generalization of Raabe's cosine transform. Several of its properties are also demonstrated.