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Error bounds for the asymptotic expansions of the Hermite polynomials
Published online by Cambridge University Press: 26 January 2022
Abstract
In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel–Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the ‘branch cut’ technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 2 , April 2023 , pp. 417 - 440
- Copyright
- Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
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