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Explicit formulas for the Bernoulli and Euler polynomials and numbers. (English) Zbl 0748.11016
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.
Reviewer: L.Skula (Brno)

11B68Bernoulli and Euler numbers and polynomials
05A19Combinatorial identities, bijective combinatorics
Full Text: DOI
[1] L. Comtet, Advanced Combinatorics (The Art of Finite and Infinite Expansions), D. Reidel Publishing Company, Dordrecht-Holland/Boston-USA, 1974. · Zbl 0283.05001
[2] H. Alzer, Ein Duplikationstheorem für die Bernoullischen Polynome, Mitt. Math. Ges. Hamburg11 (1987), 469--471.
[3] K. Dilcher, Multiplikationstheoreme für die Bernoullischen Polynome und explizite Darstellungen der Bernoullischen Zahlen, Abh. Math. Sem. Univ. Hamburg59 (1989), 143--156. · Zbl 0712.11015 · doi:10.1007/BF02942325
[4] H.W. Gould, Combinatorial Identities, Revised Ed., Morgantown Printing and Binding Co., Morgantown, WV-USA, 1972. · Zbl 0263.05013