More explicit formulas for Bernoulli and Euler numbers
نویسندگان
چکیده
منابع مشابه
Explicit Formulas for Bernoulli and Euler Numbers
Explicit and recursive formulas for Bernoulli and Euler numbers are derived from the Faá di Bruno formula for the higher derivatives of a composite function. Along the way we prove a result about composite generating functions which can be systematically used to derive such identities.
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Let [x] be the integral part of x. Let p > 5 be a prime. In the paper we mainly determine P[p/4] x=1 1 xk (mod p2), p−1 [p/4] (mod p3), Pp−1 k=1 2 k (mod p3) and Pp−1 k=1 2 k2 (mod p2) in terms of Euler and Bernoulli numbers. For example, we have
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The values at x=0 are called Bernoulli and Euler numbers of order w; when w=1, the polynomials or numbers are called ordinary. When x=0 or w=1, we often suppress that part of the notation; e.g., B (w) n denotes B n (0), En(x) denotes E (1) n (x), and Bn denotes B (1) n (0). These numbers have been extensively studied and many congruences for them are known. Among the most important results are ...
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ژورنال
عنوان ژورنال: Involve, a Journal of Mathematics
سال: 2015
ISSN: 1944-4184,1944-4176
DOI: 10.2140/involve.2015.8.275