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Zbl 0609.10008
Dilcher, Karl
Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. (English)
[J] J. Approximation Theory 49, 321-330 (1987). ISSN 0021-9045

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 ${\Bbb C}$ to the sine or cosine functions when the indices of the polynomicals tend to infinity. \par These results generalize earlier ones of the author [C. R. Math. Acad. Sci., Soc. R. Can. 6, 273--278 (1984; Zbl 0558.10012)]. \par \{Reviewer's remark: (i) For the first time some of the cited formulas were obtained by {\it 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.\}
[I.Sh.Slavutskij]

MSC 2000:: *11B68 Bernoulli numbers, etc.
41A80 Remainders in approximation formulas

Keywords: Bernoulli numbers; generalized Bernoulli numbers; generalized Bernoulli polynomials; Euler polynomials; truncated MacLaurin series

Citations: Zbl 0558.10012; Zbl 0080.03002

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