Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials
type: Item
instance of: scholarly article
title: Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials
zbMATH Open document ID: 0609.10008
DOI: 10.1016/0021-9045(87)90071-2
author: Karl Dilcher
published in: Journal of Approximation Theory
publication date: 1987-01-01T00:00:00Z
review text: The author considers some interesting problems of the asymptotic behaviour of Bernoulli, Euler and generalized (in the Berger-Leopoldt sense) Bernoulli polynomials. In particular, it is shown that the polynomials in question can be approximated by the truncated MacLaurin series for sine and cosine (with explicit error bounds) and that the sequences of polynomials under consideration converge uniformly on compact subsets of \(\mathbb C\) to the sine or cosine functions when the indices of the polynomicals tend to infinity. These results generalize earlier ones of the author [C. R. Math. Acad. Sci., Soc. R. Can. 6, 273--278 (1984; Zbl 0558.10012)]. \{Reviewer's remark: (i) For the first time some of the cited formulas were obtained by \textit{H. W. Leopoldt} [Abh. Math. Semin. Univ. Hamb. 22, 131--140 (1958; Zbl 0080.03002)] but this reference is missing; (ii) The remark to Theorem 2 contains a misprint.\}
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 41A80
zbMATH DE Number: 3985310
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: generalized Bernoulli numbers
zbMATH Keywords: generalized Bernoulli polynomials
zbMATH Keywords: Euler polynomials
zbMATH Keywords: truncated MacLaurin series
reviewed by: Ilya Sh. Slavutskii
MaRDI profile type: Publication
full work available at URL: 0021-9045(87)90071-2
OpenAlex ID: W2156692423
cites work: Q3222919
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description: scientific article; zbMATH DE number 3985310