Gabor Szegö: Collected Papers
About: This article is published in American Mathematical Monthly.The article was published on 1982-01-01. It has received 21 citations till now.
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01 Jan 1985
TL;DR: The classical orthogonal polynomials have been defined in this paper, and a number of orthogonality relations for some of the classical polynomial classes have been established.
Abstract: There have been a number of definitions of the classical orthogonal polynomials, but each definition has left out some important orthogonal polynomials which have enough nice properties to justify including them in the category of classical orthogonal polynomials. We summarize some of the previous work on classical orthogonal polynomials, state our definition, and give a few new orthogonality relations for some of the classical orthogonal polynomials.
532 citations
TL;DR: In this article, the Okamoto τ-function theory of PIV and PII has been applied to the evaluation of the largest eigenvalue of the finite and scaled infinite GUE.
Abstract: Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of \(\), where \(\) for \(\) and \(\) otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of \(\). Of particular interest are \(\) and FN(λ;2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities \(\) and FN(λ;a) for the other classical matrix ensembles.
249 citations
TL;DR: In recent years, there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory as mentioned in this paper, and some of the major achievements and a number of applications.
Abstract: In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory We review the history of these connections, some of the major achievements and a number of applications
119 citations
TL;DR: In this article, the authors present general results concerning so-called biorthogonal polynomials of R"I"I type introduced by M. Ismail and D. Masson.
90 citations
TL;DR: In this article, a new class of polynomial Vandermonde-like matrices is introduced, which generalizes the polynomials studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and solving the associated linear systems.
52 citations