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Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. (English) Zbl 0609.10008

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 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 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.}

Reviewer: I.Sh.Slavutskij
MSC:
11B68Bernoulli and Euler numbers and polynomials
41A80Remainders in approximation formulas