This book originated as Special Functions: A Graduate Text. The current version is considerably enlarged: the number of chapters devoted to orthogonal polynomials has increased from two to four; Meijer G-functions and Painlevé transcendents are now treated.

As we noted in the earlier book, the subject of special functions lacks a precise delineation, but it has a long and distinguished history. The remarks at the end of each chapter discuss the history, with numerous references and suggestions for further reading.

This book covers most of the standard topics and some that are less standard. We have tried to provide context for the discussion by emphasizing unifying ideas. The text and the problems provide proofs or proof outlines for nearly all the results and formulas.

We have also tried to keep the prerequisites to a minimum: a reasonable familiarity with power series and integrals, convergence, and the like. Some proofs rely on the basics of complex function theory, which are reviewed in the first appendix. Some familiarity with Hilbert space ideas, in the L2 framework, is useful. The chapters on elliptic functions and on Painlevé transcendents rely more heavily than the rest of the book on concepts from complex analysis. The second appendix contains a quick development of basic results from Fourier analysis, including the Mellin transform.

The first chapter provides a general context for the discussion of the linear theory, especially in connection with special properties of the hypergeometric and confluent hypergeometric equations. Chapter 2 treats the gamma and beta functions at some length, with an introduction to the Riemann zeta function. Chapter 3 covers the relevant material from the theory of ordinary differential equations, including a characterization of the classical polynomials as eigenfunctions, and a discussion of separation of variables for equations involving the Laplacian.

The next four chapters are concerned with orthogonal polynomials on a real interval. Chapter 4 introduces the general theory, including three-term recurrence relations, Padé approximants, continued fractions, and Favard's theorem. The classical polynomials (Hermite, Laguerre, Jacobi) are treated in detail in Chapter 5, including asymptotic distribution of zeros. Chapter 6 introduces finite difference analogues of the classification theorem, yielding the classical discrete polynomials as well as neoclassical versions and the Askey scheme. Two methods of obtaining asymptotic results are presented in Chapter 7. In particular, the Riemann–Hilbert method is carried through for Hermite polynomials.