Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Preliminaries
- 2 General Orthogonal Polynomials
- 3 Jacobi and Related Polynomials
- 4 Recursively Defined Polynomials
- 5 Wilson and Related Polynomials
- 6 Discrete Orthogonal Polynomials
- 7 Some q-Orthogonal Polynomials
- 8 The Askey–Wilson Family of Polynomials
- 9 Orthogonal Polynomials on the Unit Circle
- 10 Zeros of Orthogonal Polynomials
- 11 The Moment Problem
- 12 Matrix-Valued Orthogonal Polynomials and Differential Equations
- 13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
- References
- Index
4 - Recursively Defined Polynomials
Published online by Cambridge University Press: 14 September 2020
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Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- 1 Preliminaries
- 2 General Orthogonal Polynomials
- 3 Jacobi and Related Polynomials
- 4 Recursively Defined Polynomials
- 5 Wilson and Related Polynomials
- 6 Discrete Orthogonal Polynomials
- 7 Some q-Orthogonal Polynomials
- 8 The Askey–Wilson Family of Polynomials
- 9 Orthogonal Polynomials on the Unit Circle
- 10 Zeros of Orthogonal Polynomials
- 11 The Moment Problem
- 12 Matrix-Valued Orthogonal Polynomials and Differential Equations
- 13 Some Families of Matrix-Valued Jacobi Orthogonal Polynomials
- References
- Index
Summary
A birth and death process is a stationary Markov process whose states are labeled by nonnegative integers and whose transition probabilities
pm,n(t)=Pr{X(t)=n|X(0)=m} (4.1.1)
satisfy certain conditions as t→0+:
{pn,n+1(t)=λnt=o(t),pn,n−1(t)=μnt+o(t),pn,n(t)=1−(λn+μn)t+o(t),pn,m(t)=o(t),|m−n|>1.
It is assumed that
λn>0,n≥0 and μ0≥0,μn>0,n>0. (4.1.2)
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- Information
- Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 100 - 118Publisher: Cambridge University PressPrint publication year: 2020