Asymptotic expansions for solutions of smooth recurrence equations

Jha, Shing-Whu; Máté, Attila; Nevai, Paul

doi:10.1090/S0002-9939-1990-1014646-7

Asymptotic expansions for solutions of smooth recurrence equations
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by Shing-Whu Jha, Attila Máté and Paul Nevai PDF

Proc. Amer. Math. Soc. 110 (1990), 365-370 Request permission

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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