## An explicit formula for the generalized Bernoulli polynomials.(English)Zbl 0621.33008

The authors prove a new explicit formula for the generalized Bernoulli polynomials. The main result, expressing these polynomials in terms of the Gaussian hypergeometric function, provides an interesting extension of a representation for the generalized Bernoulli numbers given recently by P. G. Todorov [C. R. Acad. Sci., Paris, Sér. I 301, 665-666 (1985; Zbl 0606.10008)]. This transition and several other connections are also indicated.

### MSC:

 33C05 Classical hypergeometric functions, $${}_2F_1$$ 11B39 Fibonacci and Lucas numbers and polynomials and generalizations

### Keywords:

Bernoulli polynomials; Bernoulli numbers

Zbl 0606.10008
Full Text:

### References:

 [1] Appell, P; de Fériet, J.Kampé, Fonctions hypergéométriques et hypersphériques: polynômes d’Hermite, (1926), Gauthier-Villars Paris · JFM 52.0361.13 [2] Bailey, W.N, Generalized hypergeometric series, (1935), Cambridge Univ. Press Cambridge/London/New York · Zbl 0011.02303 [3] Comtet, L, Advanced combinatorics: the art of finite and infinite expansions, (1974), Reidel Dordrecht/Boston, (Translated from the French by J.W. Nienhuys) [4] Gould, H.W, Explicit formulas for Bernoulli numbers, Amer. math. monthly, 79, 44-51, (1972) · Zbl 0227.10010 [5] Srivastava, H.M; Lavoie, J.-L; Tremblay, R, A class of addition theorems, Canad. math. bull., 26, 438-445, (1983) · Zbl 0504.33007 [6] Todorov, P.G, Une formule simple explicite des nombres de Bernoulli généralisés, C.R. acad. sci. Paris Sér. I math., 301, 665-666, (1985) · Zbl 0606.10008
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