Identities for Bernoulli polynomials and Bernoulli numbers

摘要

We prove that if m and ν are integers with 0 ≤ ν ≤ m and x is a real number, then (1) m-1∑k=0 k+m odd (mk) (k+m v) B_(k+m-v)(x)=1/2 m∑j=0 (-1)~(j+m)(mj)(j+m-1 v)(j+m)x~(j+m-v-1), where Bn(x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers Bn. Among others, we obtain (2) m-1∑k=0 k+m odd(mk)(k+m v)(k+m-v j)B_(k+m-v-j)=0 (0≤j≤m-2-v). This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, ν = 0 and j = 3, ν = 0, respectively.