By W. A. Al-Salam (auth.), Paul Nevai (eds.)

This quantity comprises the complaints of the NATO complicated learn Institute on "Orthogonal Polynomials and Their purposes" held on the Ohio kingdom collage in Columbus, Ohio, U.S.A. among may possibly 22,1989 and June 3,1989. The complicated research Institute essentially focused on these points of the speculation and perform of orthogonal polynomials which surfaced long ago decade while the idea of orthogonal polynomials began to adventure an unheard of development. This development all started with Richard Askey's nearby Confer ence Lectures on "Orthogonal Polynomials and precise services" in 1975, and next discoveries ended in a considerable revaluation of one's perceptions as to the character of orthogonal polynomials and their applicability. the hot acclaim for orthogonal polynomials is just in part as a result of Louis de Branges's answer of the Bieberbach conjecture which makes use of an inequality of Askey and Gasper on Jacobi polynomials. the most cause lies of their large applicability in components reminiscent of Pade approximations, persisted fractions, Tauberian theorems, numerical research, chance conception, mathematical facts, scattering idea, nuclear physics, good nation physics, electronic sign processing, electric engineering, theoretical chemistry and so on. This was once emphasised and convincingly proven throughout the shows via either the vital audio system and the invited exact teachers. the most matters of our complicated examine Institute incorporated complicated orthogonal polynomials, sign processing, the recursion process, combinatorial interpretations of orthogonal polynomials, computational difficulties, capability conception, Pade approximations, Julia units, distinctive services, quantum teams, weighted approximations, orthogonal polynomials linked to root structures, matrix orthogonal polynomials, operator idea and crew representations.

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Those complaints include lectures given on the N. A. T. O. complicated examine Institute entitled "Scattering conception in arithmetic and Physics" held in Denver, Colorado, June 11-29, 1973. we've got assembled the most sequence of lectures and a few provided by means of different contributors that appeared certainly to counterpoint them.

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3. ASKEY-WILSON POLYNOMIALS AND THE CLASSIFICATION PROBLEM FOR P - AND Q -POLYNOMIAL ASSOCIATION SCHEMES In the previous section, we remarked that H(d,q) and J(v,d) have both P-polynomial and Q -polynomial structures. Are there many association schemes which have both structures? We call such a symmetric association scheme a P - and Q -polynomial association 32 scheme. We will discuss the classification problem of these schemes. This problem is not just one of the most important and interesting problems in Algebraic Combinatorics but also very interesting for its connection with orthogonal polynomials.