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

Author: Ken-iti Sato
Journal: Trans. Amer. Math. Soc. 309 (1988), 375-390
MSC: Primary 60J80; Secondary 26C10, 60E07
MathSciNet review: 957077
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Abstract: Zeros of the following system of polynomials are considered:

$\displaystyle \left\{ \begin{gathered}{P_0}(x) = 1, \hfill \\ {P_1}(x) = {B_0} ... ..._{n - 1}}(x)\quad {\text{for}}\;n \geqslant 1. \hfill \\ \end{gathered} \right.$

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.

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Keywords: Zeros of polynomials, sojourn time distribution, birth-and-death process, unimodal, infinitely divisible
Article copyright: © Copyright 1988 American Mathematical Society

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