Iterated beta integrals

TL;DR

This paper introduces iterated beta integrals on the universal abelian cover of the punctured projective line, unifying hyperlogarithms and beta integrals, and reveals their invariance properties that generate known and new identities in multiple zeta and omega values.

Contribution

It develops a new class of integrals called iterated beta integrals, establishing their properties and invariance, which unify existing functions and produce both known and novel relations among special values.

Findings

Reveals invariance under parameter translation generating identities

Recovers classical identities like Zagier's 2-3-2 formula

Proves a Galois descent phenomenon for multiple omega values

Abstract

We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental properties. We establish various analytic properties of these integrals with respect to both the exponent parameters and the main variables. Their key feature is invariance under simultaneous translation of the exponent parameters, which generates relations between integrals over possibly different coverings. This mechanism recovers notable identities for multiple zeta values and variants -- including Zagier's 2-3-2 formula, Murakami's $t$-value analogue, Charlton's $t$-value analogue, Zhao's $2$-$1$ formula, and Ohno's relation -- and also yields new relations, such as a proof of a Galois descent phenomenon for multiple omega values.