Shell Polynomials and Dual Birth-Death Processes

TL;DR

This paper explores the mathematical relationships between birth-death processes, orthogonal polynomials, and measures solving moment problems, providing new insights into duality and process similarity.

Contribution

It offers a novel interpretation of duality in birth-death processes through shell polynomials and proposes a modified concept of process similarity.

Findings

Enhanced understanding of duality in birth-death processes

New interpretation linking shell polynomials to process duality

Proposed modifications to process similarity concepts

Abstract

This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these relations, revealed to a large extent by Karlin and McGregor, we investigate a duality concept for birth-death processes introduced by Karlin and McGregor and its interpretation in the context of shell polynomials and the corresponding orthogonal polynomials. This interpretation leads to increased insight in duality, while it suggests a modification of the concept of similarity for birth-death processes.