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We study Markov measures and p-adic random walks with the use of states on the Cuntz algebras Op. Via the Gelfand–Naimark–Segal construction, these come from families of representations of Op. We prove that these representations reflect selfsimilarity especially well. In this paper, we consider a Cuntz–Krieger type algebra where the adjacency matrix depends on a parameter q ( q=1 is the case of Cuntz–Krieger algebra). This is an ongoing work generalizing a construction of certain measures associated to random walks on graphs.

Using standard transformation and summation formulas for basic hypergeometric series we obtain an explicit polynomial form of the $q$-analogue of the $\text{9-}\,j$ symbols, introduced by the author in a recent publication. We also consider a limiting case in which the $\text{9-}\,j$ symbol factors into two Hahn polynomials. The same factorization occurs in another limit case of the corresponding $q$-analogue.

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STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

An Explicit Polynomial Expression for a q-Analogue of the 9- j Symbols

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STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

An Explicit Polynomial Expression for a q-Analogue of the 9- j Symbols