### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 049, 15 pages      arXiv:1512.07104      https://doi.org/10.3842/SIGMA.2016.049
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Shell Polynomials and Dual Birth-Death Processes

Erik A. van Doorn
Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Received January 02, 2016, in final form May 14, 2016; Published online May 18, 2016

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.

Key words: orthogonal polynomials; birth-death processes; Stieltjes moment problem; shell polynomials; dual birth-death processes; similar birth-death processes.

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