Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes

TL;DR

This paper develops multi-indexed orthogonal polynomials for discrete variables across eight types and uses them to construct exactly solvable birth and death processes, including their discrete-time Markov chain counterparts.

Contribution

It introduces new multi-indexed orthogonal polynomials for discrete variables and applies them to exactly solve birth and death processes, expanding the class of solvable stochastic models.

Findings

Constructed exactly solvable continuous-time birth and death processes.

Derived finite-type discrete-time Markov chain models.

Extended the framework to multiple polynomial types including $q$-analogues.

Abstract

We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)($q$-)Hahn, three kinds of $q$-Krawtchouk and $q$-Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah, $q$-Racah, Meixner, little $q$-Jacobi and little $q$-Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.