On a Seminal Paper by Karlin and McGregor

TL;DR

This paper extends the spectral method analysis for birth-and-death processes pioneered by Karlin and McGregor, providing a more complete understanding and demonstrating applications of their powerful approach.

Contribution

It completes the spectral analysis of birth-and-death processes initiated by Karlin and McGregor, offering new insights and applications.

Findings

Extended the spectral expression for transition probabilities

Provided new applications of spectral methods in birth-and-death processes

Enhanced understanding of spectral techniques in stochastic processes

Abstract

The use of spectral methods to study birth-and-death processes was pioneered by S. Karlin and J. McGregor. Their expression for the transition probabilities was made explicit by them in a few cases. Here we complete their analysis and indicate a few applications of their very powerful method.