Book contents
- Frontmatter
- Contents
- List of Contributors
- Preface
- 1 General Overview of Multivariable Special Functions
- 2 Orthogonal Polynomials of Several Variables
- 3 Appell and Lauricella Hypergeometric Functions
- 4 A-Hypergeometric Functions
- 5 Hypergeometric and Basic Hypergeometric Series and Integrals Associated with Root Systems
- 6 Elliptic Hypergeometric Functions Associated with Root Systems
- 7 Dunkl Operators and Related Special Functions
- 8 Jacobi Polynomials and Hypergeometric Functions Associated with Root Systems
- 9 Macdonald–Koornwinder Polynomials
- 10 Combinatorial Aspects of Macdonald and Related Polynomials
- 11 Knizhnik–Zamolodchikov-Type Equations, Selberg Integrals and Related Special Functions
- 12 9 j-Coefficients and Higher
- Index
3 - Appell and Lauricella Hypergeometric Functions
Published online by Cambridge University Press: 30 September 2020
- Frontmatter
- Contents
- List of Contributors
- Preface
- 1 General Overview of Multivariable Special Functions
- 2 Orthogonal Polynomials of Several Variables
- 3 Appell and Lauricella Hypergeometric Functions
- 4 A-Hypergeometric Functions
- 5 Hypergeometric and Basic Hypergeometric Series and Integrals Associated with Root Systems
- 6 Elliptic Hypergeometric Functions Associated with Root Systems
- 7 Dunkl Operators and Related Special Functions
- 8 Jacobi Polynomials and Hypergeometric Functions Associated with Root Systems
- 9 Macdonald–Koornwinder Polynomials
- 10 Combinatorial Aspects of Macdonald and Related Polynomials
- 11 Knizhnik–Zamolodchikov-Type Equations, Selberg Integrals and Related Special Functions
- 12 9 j-Coefficients and Higher
- Index
Summary
Appell introduced four kinds of hypergeometric series in two variables as extensions of the hypergeometric series F(a,b,c;x), and Lauricella generalized them to hypergeometric series in m variables, and they considered systems of partial differential equations satisfied by them. In this chapter, we give definitions of Appell’s and Lauricella’s hypergeometric series and state their fundamental properties such as domains of convergence, integral representations, systems of partial differential equations, fundamental systems of solutions, and transformation formulas. We define the rank and the singular locus of a system of partial differential equations, and list them for Appell’s and Lauricella’s systems. We describe Pfaffian systems, contiguity relations, monodromy representations and twisted period relations for the systems. We give their explicit forms for Lauricella’s E_D, which is the simplest among Lauricella’s systems. We also mention the uniformization of the complement of the singular locus of E_D by the projectivization of its fundamental system of solutions.
Keywords
- Type
- Chapter
- Information
- Encyclopedia of Special Functions: The Askey-Bateman Project , pp. 79 - 100Publisher: Cambridge University PressPrint publication year: 2020
- 2
- Cited by