Some cubic birth and death processes and their related orthogonal polynomials

TL;DR

This paper analyzes specific cubic birth and death processes through their orthogonal polynomials, deriving generating functions, Nevanlinna matrices, and asymptotic behaviors, contributing to the understanding of indeterminate moment problems.

Contribution

It introduces new classes of orthogonal polynomials related to cubic birth and death processes and derives their generating functions and Nevanlinna matrices.

Findings

Generated explicit formulas for the polynomials' generating functions

Computed Nevanlinna matrices for the polynomial classes

Analyzed asymptotic behavior of the matrices in the complex plane

Abstract

The orthogonal polynomials with recurrence relation \[(\la\_n+\mu\_n-z) F\_n(z)=\mu\_{n+1} F\_{n+1}(z)+\la\_{n-1} F\_{n-1}(z)\] with two kinds of cubic transition rates $\la\_n$ and $\mu\_n,$ corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.