Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and ({nu}) are integers with ({0 leq nu leq m}) and x is a real number, then

1. $$sum_{k=0 atop k+m , , odd}^{m-1} {m choose k}{k+m choose nu} B_{k+m-nu}(x) = frac{1}{2} sum_{j=0}^m (-1)^{j+m} {m choose j}{j+m-1 choose nu} (j+m) x^{j+m-nu-1},$$ 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
2. $$sum_{k=0 atop k+m , , odd}^{m -1} {m choose k}{k+m choose nu} {k+m-nu choose j}B_{k+m-nu-j} =0 quad{(0 leq j leq m-2-nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, ({nu=0}) and j = 3, ({nu=0}) , respectively.