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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights

Authors: Attila Máté, Paul Nevai and Thomas Zaslavsky
Journal: Trans. Amer. Math. Soc. 287 (1985), 495-505
MSC: Primary 42C05; Secondary 05A15
MathSciNet review: 768722
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Abstract: Let $ {p_n}(x) = {\gamma _n}{x^n} + \cdots $ denote the $ n$th polynomial orthonormal with respect to the weight $ \exp ( - {x^\beta }/\beta )$ where $ \beta > 0$ is an even integer. G. Freud conjectured and Al. Magnus proved that, writing $ {a_n} = {\gamma _{n - 1}}/{\gamma _n}$, the expression $ {a_n}{n^{ - 1/\beta }}$ has a limit as $ n \to \infty $. It is shown that this expression has an asymptotic expansion in terms of negative even powers of $ n$. In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.

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Keywords: Asymptotic expansion, combinatorial enumerations, combinatorial identities, Freud's conjecture, Jensen's identity, orthogonal polynomials
Article copyright: © Copyright 1985 American Mathematical Society

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