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Asymptotic expansions for solutions of smooth recurrence equations


Authors: Shing-Whu Jha, Attila Máté and Paul Nevai
Journal: Proc. Amer. Math. Soc. 110 (1990), 365-370
MSC: Primary 33C45; Secondary 39A10, 41A60
DOI: https://doi.org/10.1090/S0002-9939-1990-1014646-7
MathSciNet review: 1014646
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Abstract: Let $ \langle {y_n}:n \geq 1\rangle $ be a convergent sequence of reals, where for each $ n$ the tuple $ \langle {y_n},{y_{n + 1}}, \ldots ,{y_n} + k,1/n\rangle $ satisfies one of $ r$ equations, depending on the residue class of $ n(\bmod r)$, for some given $ k$ and $ r$ . Assume these equations are smooth, they have the same gradient in the first $ k + 1$ variables, and this gradient satisfies a certain nonmodularity condition. We then show that $ {y_n}$ has $ r$ asymptotic expansions, depending on the residue class of $ n(\bmod r)$, in terms of powers of $ 1/n$. This result enables us to discuss the asymptotic behavior of the recurrence coefficients associated with certain orthogonal polynomials. A key ingredient in the proof of the main result is a lemma involving exponential sums.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1014646-7
Keywords: Asymptotic expansions, exponential sums, orthogonal polynomials, recurrence equations
Article copyright: © Copyright 1990 American Mathematical Society

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