A new deformation of multiple zeta value

Abstract We introduce a new deformation of multiple zeta value (MZV). It has one parameter $$\omega $$ ω satisfying $$0<\omega <2$$ 0 < ω < 2 and recovers MZV in the limit as $$\omega \rightarrow +0$$ ω → + 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.