A new deformation of multiple zeta value

TL;DR

This paper introduces a new one-parameter deformation of multiple zeta values (MZVs) that generalizes MZVs and satisfies key algebraic relations similar to those of $q$-analogues, using a multiple integral involving hyperbolic gamma functions.

Contribution

It presents a novel deformation of MZVs parameterized by $ extomega$, proving it satisfies double shuffle and extended double Ohno relations.

Findings

Deformation recovers MZV as $ extomega o 0$

Satisfies double shuffle relations

Proves extended double Ohno relations

Abstract

We introduce a new deformation of multiple zeta value (MZV). It has one parameter $\omega$ satisfying $0<\omega<2$ and recovers MZV in the limit as $\omega \to +0$. It is defined in the same algebraic framework as a $q$-analogue of multiple zeta value ($q$MZV) by using a multiple integral. We prove that our deformed multiple zeta value satisfies the double shuffle relations which are satisfied by $q$MZVs. We also prove the extended double Ohno relations, which are proved for ($q$)MZVs by Hirose, Sato and Seki, by using a multiple integral whose integrand contains the hyperbolic gamma function due to Ruijsenaars.