Basel-Boston: Birkhäuser. XVIII, 427 p.; DM 198.00 (1988).

For a review see the second edition (1984;

Zbl 0567.33001). Preface to the American edition: ”With students of physics chiefly in mind, we have collected the material on special functions that is most important in mathematical physics and quantum mechanics. We have not attempted to provide the most extensive collection possible of information about special functions, but have set ourselves the task of finding an exposition which, based on a unified approach, ensures the possibility of applying the theory in other natural sciences, since it provides a simple and effective method for the independent solution of problems that arise in practice in physics, engineering and mathematics. For the American edition we have been able to improve a number of proofs; in particular we have given a new proof of the basic theorem (§ 3). This is the fundamental theorem of the book; it has now been extended to cover difference equations of hypergeometric type (§§ 12, 13). Several sections have been simplified and contain new material. We believe that this is the first time that the theory of classical orthogonal polynomials of a discrete variable on both uniform and nonuniform lattices has been given such a coherent presentation, together with its various applications in physics.” From the translator’s preface: ”For this translation, the authors have substantially rearranged and amplified the material, particularly in § 13, where they discuss recent work on the connection between the Hahn polynomials and the Clebsch-Gordan coefficients, and between the Wigner $6j$-symbols and the Racah polynomials. They have also added (§ 27, Parts 2 and 3) applications of orthogonal polynomials of a discrete variable to the compression of information, and of Bessel functions to laser sounding of the atmosphere.”