Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.

Table of Contents:
1. General overview of multivariable special functions Tom H. Koornwinder and Jasper V. Stokman; 2. Orthogonal polynomials of several variables Yuan Xu; 3. Appell and Lauricella hypergeometric functions Keiji Matsumoto; 4. A-Hypergeometric functions Nobuki Takayama; 5. Hypergeometric and basic hypergeometric series and integrals associated with root systems Michael J. Schlosser; 6. Elliptic hypergeometric functions associated with root systems Hjalmar Rosengren and S. Ole Warnaar; 7. Dunkl operators and related special functions Charles F. Dunkl; 8. Jacobi polynomials and hypergeometric functions associated with root systems Gert J. Heckman and Eric M. Opdam; 9. Macdonald–Koornwinder polynomials Jasper V. Stokman; 10. Combinatorial aspects of Macdonald and related polynomials Jim Haglund; 11. Knizhnik–Zamolodchikov type equations, Selberg integrals, and related special functions Vitaly Tarasov and Alexander Varchenko; 12. 9j-Coefficients and higher Joris Van der Jeugt; Index.

About the Author :
Tom H. Koornwinder is Professor Emeritus at the University of Amsterdam. He is an expert in special functions, orthogonal polynomials, and Lie theory. He introduced the five-parameter extension of the BC-type Macdonald polynomials, which are nowadays called Koornwinder polynomials. He was co-author of the chapter on orthogonal polynomials in the Digital Library of Mathematical Functions, and is involved in its revision. Jasper V. Stokman is Professor of Mathematics at the University of Amsterdam. He is an expert in special functions, Lie theory, and integrable systems. He introduced BC-type extensions of several families of classical orthogonal polynomials, and nonpolynomial generalizations of Koornwinder polynomials. He linked multivariable special functions to harmonic analysis on quantum groups and Hecke algebras, and to statistical mechanics and analytic number theory.

Review :
'Overall the volume is a very useful addition to any research mathematician's library who works on these topics.' Manjil Pratim Saikia, zbMATH