A new deformation of multiple zeta value

We introduce a new deformation of multiple zeta value (MZV). It has one parameter satisfying 0<<2 and recovers MZV in the limit as +0. It is defined in the same algebraic framework as a q-analogue of multiple zeta value (qMZV) by using a multiple integral. We prove that our deformed multiple zeta value satisfies the double shuffle relations which are satisfied by qMZVs. 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.