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
3 - Jacobi and Related Polynomials
Published online by Cambridge University Press: 14 September 2020
Edited by
Assisted by
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
The Jacobi polynomials are defined by
Pn(α,β)(x)=(α+1)nn!F12(−n,α+β+n+1;α+1;(1−x)/2), (3.1.1)
and satisfy the orthogonality relations
∫−11Pm(α,β)(x)Pn(α,β)(x)(1−x)α(1+x)βdx=hn(α,β)δm,n, (3.1.2)
hn(α,β)=2α+β+1Γ(α+n+1)Γ(β+n+1)n!Γ(α+β+n+1)(α+β+2n+1). (3.1.3)
- Type
- Chapter
- Information
- Publisher: Cambridge University PressPrint publication year: 2020