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

TL;DR

This work extends case-(1) multi-indexed orthogonal polynomials on a discrete variable to eight types within the Askey scheme and harnesses rdQM to derive exactly solvable birth–death processes from these polynomials. It introduces deformation data and similarity-transformed Hamiltonians to construct forward/backward relations, enabling explicit spectral decompositions and stationary distributions for both continuous- and discrete-time BD processes, including repeated-time variants. A key methodological insight is using the ratio of deformed polynomials and alternative similarity transforms to ensure probability conservation while maintaining exact solvability. The results broaden the analytic toolbox linking discrete orthogonal polynomials to stochastic dynamics and offer explicit formulas for transition probabilities and long-time behavior in complex BD systems.

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.