On Berndt's summation formula
Alexander E. Patkowski
TL;DR
This paper provides a new proof of Berndt's summation formula using Mellin inversion, Muntz, and Poisson summation formulas, and explores new applications and examples.
Contribution
It introduces a novel proof technique for Berndt's summation formula and presents several new applications and examples derived from this approach.
Findings
Proof of Berndt's summation formula using classical analysis tools
Examples involving discontinuous functions demonstrating the formula's applicability
New corollaries expanding the formula's utility
Abstract
We offer a proof of a summation formula equivalent to one due to Berndt. Our proof uses the Mntz formula and the Poisson summation formula. By utilizing known properties of Mellin inversion, we give an example from a discontinuous function. Several new applications are offered as corollaries.
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Taxonomy
TopicsAdvanced Mathematical Identities · Advanced Combinatorial Mathematics · Analytic Number Theory Research