The subject of special functions is one that has no precise delineation. This book includes most of the standard topics and a few that are less standard. The subject does have a long and distinguished history, which we have tried to highlight through remarks and numerous references. The unifying ideas are easily lost in a forest of formulas. We have tried to emphasize these ideas, especially in the early chapters.

To make the book useful for self-study we have included introductory remarks for each chapter, as well as proofs, or outlines of proofs, for almost all the results. To make it a convenient reference, we have concluded each chapter with a concise summary, followed by brief remarks on the history, and references for additional reading.

We have 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 Appendix A. The necessary background from differential equations is covered in Chapter 3. Some familiarity with Hilbert space ideas, in the L2 framework, is useful but not indispensable. Chapter 11 on elliptic functions relies more heavily than the rest of the book on concepts from complex analysis. Appendix B contains a quick development of basic results from Fourier analysis.

The first-named author acknowledges the efforts of some of his research collaborators, especially Peter Greiner, Bernard Gaveau, Yakar Kannai, and Jacek Szmigielski, who managed over a period of years to convince him that special functions are not only useful but beautiful.