On a Seminal Paper by Karlin and McGregor
Mirta M. Castro, F. Alberto Gr\"unbaum

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.
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