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arXiv · 2112.09819

Ramanujan and Koshliakov Meet Abel and Plana

Bruce C. Berndt·Atul Dixit·Rajat Gupta·Alexandru Zaharescu
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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.

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Bruce C. Berndt, Atul Dixit, Rajat Gupta, Alexandru Zaharescu. 2021-12-18. Ramanujan and Koshliakov Meet Abel and Plana. https://arxiv.org/abs/2112.09819

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