Special functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. The authors provide organizing ideas, motivation, and historical background for the study and application of some important special functions. This clearly expressed and readable work can serve as a learning tool and lasting reference for students and researchers in special functions, mathematical physics, differential equations, mathematical computing, number theory, and combinatorics.

Review

'Occasionally there is published a mathematics book that one is compelled to describe as, well, let us say, special. Special Functions is certainly one of those rare books. ... this treatise ... should become a classic. Every student, user, and researcher in analysis will want to have it close at hand as she/he works.' The Mathematical Intelligencer

' ... the material is written in an excellent manner ... I recommend this book warmly as a rich source of information to everybody who is interested in 'Special Functions'.' Zentralblatt MATH

' ... this book contains a wealth of fascinating material which is presented in a user-friendly way. If you want to extend your knowledge of special functions, this is a good place to start. Even if your interests are in number theory or combinatorics, there is something for you too ... the book can be warmly recommended and should be in all good libraries.' Adam McBride, The Mathematical Gazette

' ... it comes into the range of affordable books that you want to (and probably should have on your desk'. Jean Mawhin, Bulletin of the Belgian Mathematical Society

'The book is full of beautiful and interesting formulae, as was always the case with mathematics centred around special functions. It is written in the spirit of the old masters, with mathemtics developed in terms of formulas. There are many historical comments in the book. It can be recommended as a very useful reference.' European Mathematical Society

'... full of beautiful and interesting formulae ... It can be recommended as a very useful reference.' EMS Newsletter

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