Ramanujan and Koshliakov Meet Abel and Plana

TL;DR

This paper explores the work of Koshliakov and Ramanujan on generalizations of the Abel--Plana summation formula, connecting classical results with new generalizations derived from boundary value problems and Eisenstein series.

Contribution

It introduces new generalizations of the Abel--Plana formula based on Koshliakov's boundary value problem approach and extends Ramanujan's analogues to Eisenstein series.

Findings

Koshliakov's formulas reduce to Abel--Plana as p→∞

New generalizations of Abel--Plana are rigorously formulated and proved

A broad generalization of Eisenstein series transformation is achieved

Abstract

The neglected Russian mathematician, N.~S.~Koshliakov, derived beautiful generalizations of the classical Abel--Plana summation formula through a setting arising from a boundary value problem in heat conduction. When we let the parameter $p$ in this setting tend to infinity, his formulas reduce to the classical Abel--Plana summation formula. Rigorous formulations and proofs of these summation formulas are given. In his notebooks, Ramanujan derived different analogues of the Abel--Plana summation formula. One particular example provides a vast new generalization of the classical transformation formula for Eisenstein series, which we generalize in Koshliakov's setting.