Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0748.11016
Todorov, P.G.
Explicit formulas for the Bernoulli and Euler polynomials and numbers. (English)
[J] Abh. Math. Semin. Univ. Hamb. 61, 175-180 (1991). ISSN 0025-5858; ISSN 1865-8784/e

In this paper the main result (Theorem 2) gives the following formula for the Bernoulli polynomials $B\sb n(x)$ $$(te\sp{tx}/(e\sp t-1)=\sum\sp \infty\sb{n=0}B\sb n(x)t\sp n/n!,\quad \vert t\vert<2\pi):$$ $$B\sb n(\lambda z)=\lambda\sp nB\sb n(z)+n\sum\sp n\sb{n=1}\sum\sp{\nu- 1}\sb{k=0}(-1)\sp \nu{n\choose\nu}E\sb \lambda(n,\nu,k)(k+\lambda z)\sp{n-1},$$ where $z$ is a complex number, $n\ge 1$ and $\lambda\ge 2$ are integers, and $$E\sb \lambda(n,\nu,k)=\sum\sp{\lambda- 1}\sb{j=1}\varepsilon\sb \lambda\sp{(\nu-k)j}/(1-\varepsilon\sp j\sb \lambda)\sp n,\quad\varepsilon\sb \lambda=\exp i2\pi/\lambda.$$ Furthermore the author derives (Theorem 1) twelve formulas for the Bernoulli and Euler numbers and the Bernoulli and Euler polynomials, e.g. $$B\sb n=(n/2\sp n(2\sp n-1))\sum\sp n\sb{\nu=1}\sum\sp{\nu-1}\sb{k=0}(- 1)\sp{k+1}{n\choose\nu}k\sp{n -1},\quad n\ge 1.$$ The proofs make use of the combinatorial identity of {\it H. W. Gould} [Combinatorial identities (1972; Zbl 0241.05011)] $$\sum\sp n\sb{m=k}{m-a\choose k-a}x\sp m=x\sp n\sum\sp n\sb{\nu=k}{n-a+1\choose \nu-a+1}((1-x)/x)\sp{\nu-k}$$ and the formulas of {\it H. Alzer} [Mitt. Math. Ges. Hamb. 11, 469-471 (1987; Zbl 0632.10008)] and {\it K. Dilcher} [Abh. Semin. Univ. Hamb. 59, 143- 156 (1989; Zbl 0712.11015)] for the Bernoulli and Euler polynomials.
[L.Skula (Brno)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
05A19 Combinatorial identities

Keywords: Bernoulli numbers; Gould's identity; Bernoulli polynomials; Euler numbers; Euler polynomials

Citations: Zbl 0241.05011; Zbl 0632.10008; Zbl 0712.11015

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]