Several explicit formulae for Bernoulli polynomials
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Introduction Over the years, Bernoulli numbers Bn and polynomials Bn(x) have proven to be important mathematical objects. Since they appeared on the scene (17th century), they have been of interest to many mathematicians and they have appeared in many different fields of mathematics (see the interesting document of Mazur ). Bernoulli polynomials are usually defined by means of the generating function [1, p. 48] text e t − 1 = X∞ n=0 Bn(x) t n n! (|t| < 2π). We can also find Bernoulli polynomials defined by [8, p. 367]: Bn(x) = Xn k=0 n k Bkx n−k . We mention the explicit formula [6, Vol. 8, (2.6)] Bn(x) = Xn i=0 1 i + 1 X i j=0 (−1)j i j (x + j) n , (1) since this is the known formula for Bernoulli polynomials that will be used in this paper. © 2016 Department of Mathematics, University of Osijek.