Asymptotics of Certain Sums Required in Loop Regularisation

TL;DR

This paper proves three conjectures about the asymptotic behavior of specific sums involving binomials, powers, and logarithms related to loop regularisation, extending their scope and providing full asymptotic series and exact formulas.

Contribution

It confirms three conjectures, generalizes their results, and derives comprehensive asymptotic series and exact formulas for sums used in loop regularisation.

Findings

All three conjectures are proven true.

Extended sums to more general powers.

Derived full asymptotic series and exact formulas.

Abstract

We consider the three conjectures stated in a 2003 paper of Wu, concerning the asymptotics of particular sums of products of binomials, powers and logarithms. These sums relate to the form of the regularised integrals used in loop regularisation. We show all three are true, extend them to more general powers and produce their full asymptotic series. We also extend a classical result to produce an exact formula for the sum in the last.