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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 (x)

(te tx /(e t -1)= n=0 B n (x)t n /n!,|t|<2π):
B n (λz)=λ n B n (z)+n n=1 n k=0 ν-1 (-1) ν n νE λ (n,ν,k)(k+λz) n-1 ,

where z is a complex number, n1 and λ2 are integers, and

E λ (n,ν,k)= j=1 λ-1 ε λ (ν-k)j /(1-ε λ j ) n ,ε λ =expi2π/λ·

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

B n =(n/2 n (2 n -1)) ν=1 n k=0 ν-1 (-1) k+1 n νk n-1 ,n1·

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

m=k n m-a k-ax m =x n ν=k n n-a+1 ν-a+1((1-x)/x) ν-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:
11B68Bernoulli and Euler numbers and polynomials
05A19Combinatorial 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.