A unified approach for summation formulae

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.