Arith . IDENTITIES CONCERNING BERNOULLI AND EULER POLYNOMIALS

We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If n is a positive integer, r + s + t = n and x + y + z = 1, then we have r s t x y n + s t r y z n + t r s z x n = 0 where s t x y n := n k=0 (−1) k s k t n − k B n−k (x)B k (y). It is interesting to compare this with the following property of determinants: r s t x y + s t r y z + t r s z x = 0. Our symmetric relation implies the curious identities of Miki and Matiya-sevich as well as some new ones for Bernoulli polynomials such as n k=0 n k 2 B k (x)B n−k (x) = 2 n k=0 k =n−1 n k n + k − 1 k B k (x)B n−k .