Hypergeometric Multiple Orthogonal Polynomials and Random Walks

Abstract

A. The recently found hypergeometric multiple orthogonal polynomials on the step-line by Lima and Loureiro are shown to be random walk polynomials. It is proven that the corresponding Jacobi matrix and its transpose, which are nonnegative matrices and describe higher recurrence relations, can be normalized to two stochastic matrices, dual to each other. Using the Christo el-Darboux formula on the step-line and the Poincaré theory for non-homogeneous recurrence relations it is proven that both stochastic matrices are related by transposition in the large 𝑛 limit. These random walks are beyond birth and death, as they describe a chain in where transitions to the two previous states are allowed, or in the dual to the two next states. The corresponding Karlin-McGregor representation formula is given for these new Markov chains. The regions of hypergeometric parameters where the Markov chains are recurrent or transient are given. Stochastic factorizations, in terms of pure birth and of pure death factors, for the corresponding Markov matrices of types I and II, are provided. Twelve uniform Jacobi matrices and the corresponding random walks, related to a Jacobi matrix of Toeplitz type, and theirs stochastic or semi-stochastic matrices (with sinks and sources), that describe Markov chains beyond birth and death, are found and studied. One of these uniform stochastic cases, which is a recurrent random walk, is the only hypergeometric multiple random walk having a uniform stochastic factorization. The corresponding weights, Jacobi and Markov transition matrices and sequences of type II multiple orthogonal polynomials are provided. Chain of Christo el transformations connecting the stochastic uniform tuples between them, and the semi-stochastic uniform tuples, between them, are presented. As byproduct, summations formulas at unity and three and four terms contiguous relations for the generalized hypergeometric function 𝐹 3 2 are found. Moreover, using the uniform recurrence relation and generating functions explicit expressions are found for the type I multiple orthogonal polynomials. Finally, the recent Karp-Prilepkina summation formulas, extension of the Karlsson-Minton formulas, are applied to find the corresponding summations for the generalized moments of the type II multiple orthogonal polynomials related to the remainder in the corresponding type II Hermite-Padé approximation problem.

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and random walks

“…Notice that in [2] this was taken as granted, and that in the hypergeometric case [29,3] and the Jacobi-Piñeiro in the semi-band [2,5] is also true that 𝛼 2 > 0. This motivates the following notion:…”

“…𝑁+1 of the auxiliary matrix T [𝑁] satisfy B [1] 𝑁+1 𝜆 [𝑁] 𝑛 = 𝐵 [1] 𝑁+1 𝜆 [𝑁] 𝑛 + 𝛼 2 𝐵 [2] 𝑁+1 𝜆 [𝑁] 𝑛 , B [2] 𝑁+1 (𝜆 [𝑁] 𝑛 ) = 𝐵 [2] 𝑁+1 𝜆 [𝑁] 𝑛 + 𝛼 5 𝐵 [3] 𝑁+1 𝜆 [𝑁] 𝑛 , . .…”

“…Our motivation for this work initiated in our previous research on random walks beyond birth and death and multiple orthogonal polynomials, see the papers [3,5], in where the transition matrices were Hessenberg matrices with more than three diagonals. In those papers we studied two specific cases, in [3] we dealt with the Jacobi-Piñeiro multiple orthogonal polynomials, see [22,Section 23.3.2] and in [5] the hypergeometric multiple orthogonal polynomials recently presented in [29].…”

On multiple orthogonal polynomials on the step-line

“…This paper is devoted to the study of multiple orthogonal polynomials with two weights on the step-line. In the research on the relation between Markov chains beyond birth and death processes and multiple orthogonal polynomials [9,10] we get, as a byproduct, some results that hold for families of these type I and II simple cases. Moreover, in order to better understand some of these phenomena we turned back to the definition of multiple orthogonality and find, to the best of our knowledge, some unknown facts so far.…”

“…It is also shown that the type I linear form associated to the sequence of type II multiple orthogonal polynomials {𝐵 (𝑛) } ∞ 𝑛=0 is also uniquely determined. Examples of this gauge freedom with almost uniform Jacobi matrices are obtained in [10].…”