Multivariate Meixner polynomials as Birth and Death polynomials

TL;DR

This paper constructs multivariate Meixner polynomials explicitly as Birth and Death polynomials, providing a new multivariate orthogonal polynomial family with applications to stochastic processes.

Contribution

It introduces a novel multivariate Meixner polynomial family as eigenfunctions of a specific birth and death process, extending single-variable properties to multiple dimensions.

Findings

Polynomials form a complete eigenbasis for the birth and death process.

Stationary distribution generalizes the single-variable Meixner weight.

Polynomials satisfy a multivariate difference equation and are hypergeometric functions.

Abstract

Based on the framework of Plamen Iliev, multivariate Meixner polynomials are constructed explicitly as Birth and Death polynomials. They form the complete set of eigenpolynomials of a birth and death process with the birth and death rates at population $x=(x_1,\ldots,x_n)\in\mathbb{N}_0^n$ are $B_j(x)=\bigl(\beta+\sum_{i=1}^nx_j\bigr)$ and $D_j(x)=c_j^{-1}x_j$, $0<c_j$, $j=1,\ldots,n$, $\sum_{j=1}^nc_j<1$. The corresponding stationary distribution is $(\beta)_{\sum_{j=1}^nc_j}\prod_{j=1}^n(c_j^{x_j}/x_j!)(1-\sum_{j=1}^nc_j)^\beta$, the trivial $n$-variable generalisation of the orthogonality weight of the single variable Meixner polynomials. The polynomials, depending on $n+1$ parameters ($\{c_i\}$ and $\beta$), satisfy the difference equation with the coefficients $B_j(x)$ and $D_j(x)$ $j=1,\ldots,n$, which is the straightforward generalisation of the difference equation governing the single variable Meixner polynomials. The polynomials are truncated $(n+1,2n+2)$ hypergeometric functions of Aomoto-Gelfand. The polynomials and the derivation are very similar to those of the multivariate Krawtchouk polynomials reported recently.