On zeros of a system of polynomials and application to sojourn time distributions of birth-and-death processes

Abstract:

Zeros of the following system of polynomials are considered: \[ \left \{ \begin {gathered} {P_0}(x) = 1, \hfill \\ {P_1}(x) = {B_0} - {A_0}x, \hfill \\ {P_{n + 1}}(x) = ({B_n} - {A_n}x){P_n}(x) - {C_n}{P_{n - 1}}(x)\quad {\text {for}}\;n \geqslant 1. \hfill \\ \end {gathered} \right .\] Numbers of positive and negative zeros are determined and a separation property of the zeros of ${P_m}(x)$ and ${P_n}(x)$ is proved under the condition that ${C_n} > 0$ and ${P_n}(0) > 0$ for every $n$. No condition is imposed on ${A_n}$. These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.