Multiple zeta values in $\lambda$-rings

Abstract

We propose a framework for multiple zeta values in $\lambda$-rings that unifies both classical multiple zeta values and multiple $q$-zeta values. As in the classical case, we show that the multiple zeta function defines an algebra homomorphism from the quasi-shuffle algebra to the $\lambda$-ring. We also prove a formula for multiple polylogarithms in $\lambda$-rings, similar to the iterated integral expression in the classical case. As an application, we explicitly describe the algebra of regular multiple $q$-zeta values, compute the corresponding Poincaré series, and exhibit a small spanning set.

20 pages