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 | References | Similar Articles | Additional Information

Abstract: Let be a convergent sequence of reals, where for each the tuple satisfies one of equations, depending on the residue class of , for some given and . Assume these equations are smooth, they have the same gradient in the first variables, and this gradient satisfies a certain nonmodularity condition. We then show that has asymptotic expansions, depending on the residue class of , in terms of powers of . 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