ORTHOGONAL POLYNOMIALS ON ℝ⁺ AND BIRTH-DEATH PROCESSES WITH KILLING

Abstract

The purpose of this paper is to extend some results of Karlin and McGregor's and Chihara's concerning the three-terms recurrence relation for polynomials orthogonal with respect to a measure on the nonnegative real axis. Our findings are relevant for the analysis of a type of Markov chains known as birth-death processes with killing.

Representations for the extreme zeros of orthogonal polynomials

“…Generalizing a classic result of Karlin and McGregor [7], we have shown in [6] (see also [8]) that the transition probabilities…”

Shell Polynomials and Dual Birth-Death Processes

“…(1.2) (Further results in this vein can be found in [14].) Since λ n and µ n may be interpreted as the birth rates and death rates, respectively, of a birth-death process on the nonnegative integers, we will refer to a collection of such constants as a set of birth and death rates (or a rate set, for short).…”

“…Since λ n and µ n may be interpreted as the birth rates and death rates, respectively, of a birth-death process on the nonnegative integers, we will refer to a collection of such constants as a set of birth and death rates (or a rate set, for short). More information on birth-death processes and their rates will be given in later sections, but at this stage we note that, by Karlin [14,Theorem 1.3]), we must have µ 0 = 0 unless ψ has a finite moment of order −1, that is,…”