Divergent multiple sums and integrals with constraints: a comparative study

Abstract

We investigate the meromorphic behaviour of

• multiple discrete sums of tensor products of symbols with conical constraints,
• multiple integrals of tensor products of symbols with linear constraints,

and compare their pole structures. We discuss how to extract reasonable finite parts at poles leading to “renormalised” multiple discrete sums and integrals which factorise on tensor products. The first generalise “renormalised”' multiple zeta values at non positive integers, whereas the second relate to “renormalised” Feynman integrals in physics. These lectures, which are essentially self-contained, start with a review of regularised integrals and sums in one variable.