A unified approach for summation formulae

TL;DR

This paper introduces a unifying framework for classical summation formulae in analysis, linking them through a single mother-equation and deriving new asymptotic expansions for various formulae.

Contribution

It presents a novel unified formalism that connects and extends classical summation formulae via a mother-equation, enabling new asymptotic expansion results.

Findings

Derived asymptotic expansions for Voronoi and Circle formulae

Developed a M"obius-Poisson summation formula with its asymptotic expansion

Unified various classical summation formulae within a common framework

Abstract

Summation formulae are classical tools in analysis: Taylor-MacLaurin, Euler-MacLaurin, Poisson, Vorono\"i, Circle formulae\ldots We will show how, from a single equation - referred to as the mother-equation - it is possible to unify these formulae and many others within a common formalism. Indeed, these formulae are paired up: every summation formula is associated with an asymptotic expansion. For example, the Euler-MacLaurin's formula turns out to be the asymptotic expansion associated with the Poisson's formula, the Taylor-MacLaurin's formula being of course the expansion of the function initially considered. As is generally the case in sciences, this unifying concept is also generating new results. Asymptotic expansions for Vorono\"i and Circle formulae are presented first, then we show how to develop a M\"obius-Poisson's summation formula with its Euler-M\"obius-Poisson asymptotic expansion.