Jacobi–Piñeiro Markov chains

Abstract

Given a non-negative recursion matrix describing higher order recurrence relations for mul- tiple orthogonal polynomials of type II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The Karlin–McGregor representation formula is extended to both dual Markov chains and applied to the discussion of the corresponding generating functions and first-passage distributions. Recurrent or transient character of the Markov chain is discussed. The Jacobi–Piñeiro mul- tiple orthogonal polynomials are taken as a case study of the described results. The region of parameters where the recursion matrix is non-negative is given. Moreover, two stochastic matrices, describing two dual Markov chains are given in terms of the recursion matrix and the values of the multiple orthogonal polynomials of type II and corresponding linear forms of type I at the point x = 1. The region of parameters where the Markov chains are recurrent or transient is given, and the connection between both dual Markov chains is discussed at the light of the Poincaré’s theorem.