The bilateral birth-death chain generated by the associated Jacobi polynomials

TL;DR

This paper provides a probabilistic interpretation of associated Jacobi polynomials through a bilateral birth-death chain, explores matrix factorizations and transformations, and introduces an urn model for these polynomials.

Contribution

It introduces a novel probabilistic framework for associated Jacobi polynomials, including stochastic matrix factorizations and an urn model, extending classical polynomial interpretations.

Findings

Constructed a bilateral birth-death chain from associated Jacobi polynomials.

Analyzed UL and LU stochastic factorizations of the transition matrix.

Developed an urn model representing the polynomials on the integers.

Abstract

We give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three-term recurrence relation for the classical Jacobi polynomials by shifting the integer index $n$ by a real number $t$. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix which can be interpreted as the one-step transition probability matrix of a discrete-time bilateral birth-death chain with state space on $\mathbb{Z}$. We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.