QBD processes associated with Jacobi-Koornwinder bivariate polynomials and urn models

TL;DR

This paper explores QBD processes linked to Jacobi-Koornwinder bivariate polynomials, providing explicit recurrence relations and an urn model for a specific case, advancing understanding of their structure and applications.

Contribution

It explicitly computes recurrence coefficients for QBD processes related to Jacobi-Koornwinder polynomials and introduces an urn model for a special case.

Findings

Explicit recurrence coefficients derived

Conditions for discrete-time QBD processes established

Urn model constructed for a specific case

Abstract

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi-Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.