Markov measures and extended zeta functions

In this paper we study a family of representations of the Cuntz algebras O p where p is a prime. These algebras are built on generators and relations. They are C ∗ -algebras and their representations are a part of non-commutative harmonic analysis. Starting with specific generators and relations we pass to an ambient C ∗ -algebra, for example in one of the Cuntz-algebras. Our representations are motivated by the study of frequency bands in signal processing: We construct induced measures attached to those representations which turned out to be related to a class of zeta functions. For a particular case those measures give rise to a class of Markov measures and q -Bernoulli polynomials. Our approach is amenable to applications in problems from dynamics and mathematical physics: We introduce a deformation parameter q , and an associated family of q -relations where the number q is a “quantum-deformation,” and also a parameter in a scale of (Riemann-Ruelle) zeta functions. Our representations are used in turn in a derivation of formulas for this q -zeta function.