Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and v are integers with and 0 <= v <= m and x is a real number, then [GRAPHICS] [GRAPHICS] where B (n) (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B (n) . Among others, we obtain [GRAPHICS] This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1,v=0 and j = 3, v=0, respectively.