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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 ${B}_{n}\left(x\right)$

$\left(t{e}^{tx}/\left({e}^{t}-1\right)=\sum _{n=0}^{\infty }{B}_{n}\left(x\right){t}^{n}/n!,\phantom{\rule{1.em}{0ex}}|t|<2\pi \right):$
${B}_{n}\left(\lambda z\right)={\lambda }^{n}{B}_{n}\left(z\right)+n\sum _{n=1}^{n}\sum _{k=0}^{\nu -1}{\left(-1\right)}^{\nu }\left(\genfrac{}{}{0pt}{}{n}{\nu }\right){E}_{\lambda }\left(n,\nu ,k\right){\left(k+\lambda z\right)}^{n-1},$

where $z$ is a complex number, $n\ge 1$ and $\lambda \ge 2$ are integers, and

${E}_{\lambda }\left(n,\nu ,k\right)=\sum _{j=1}^{\lambda -1}{\epsilon }_{\lambda }^{\left(\nu -k\right)j}/{\left(1-{\epsilon }_{\lambda }^{j}\right)}^{n},\phantom{\rule{1.em}{0ex}}{\epsilon }_{\lambda }=expi2\pi /\lambda ·$

Furthermore the author derives (Theorem 1) twelve formulas for the Bernoulli and Euler numbers and the Bernoulli and Euler polynomials, e.g.

${B}_{n}=\left(n/{2}^{n}\left({2}^{n}-1\right)\right)\sum _{\nu =1}^{n}\sum _{k=0}^{\nu -1}{\left(-1\right)}^{k+1}\left(\genfrac{}{}{0pt}{}{n}{\nu }\right){k}^{n-1},\phantom{\rule{1.em}{0ex}}n\ge 1·$

The proofs make use of the combinatorial identity of H. W. Gould [Combinatorial identities (1972; Zbl 0241.05011)]

$\sum _{m=k}^{n}\left(\genfrac{}{}{0pt}{}{m-a}{k-a}\right){x}^{m}={x}^{n}\sum _{\nu =k}^{n}\left(\genfrac{}{}{0pt}{}{n-a+1}{\nu -a+1}\right){\left(\left(1-x\right)/x\right)}^{\nu -k}$

and the formulas of H. Alzer [Mitt. Math. Ges. Hamb. 11, 469-471 (1987; Zbl 0632.10008)] and K. Dilcher [Abh. Semin. Univ. Hamb. 59, 143- 156 (1989; Zbl 0712.11015)] for the Bernoulli and Euler polynomials.

Reviewer: L.Skula (Brno)
##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 05A19 Combinatorial identities, bijective combinatorics
##### References:
 [1] L. Comtet, Advanced Combinatorics (The Art of Finite and Infinite Expansions), D. Reidel Publishing Company, Dordrecht-Holland/Boston-USA, 1974. [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.