Some results on sums of products of Bernoulli polynomials and Euler polynomials

In this paper, by the methods of partial fraction decomposition and generating functions, we establish an explicit expression for sums of products of l Bernoulli polynomials and n?l Euler polynomials, i.e., for sums $$S_n^{(k)}(y;l,k-l):= sum_{substack{j_1+cdots+j_k=nj_1,dots,j_kgeq0}} binom {n}{j_1,dots,j_k} B_{j_1}(x_1)cdots B_{j_l}(x_l)E_{j_{l+1}}(x_{l+1}) cdots E_{j_k}(x_k). $$ This result is then used to deal with various other types of sums of products of Bernoulli polynomials and Euler polynomials. Some of them are expressed in terms of $S_{n}^{(k)}(y;l,k-l)$ and can be computed directly, while the others satisfy certain recurrences and can be determined recursively. As a consequence, many known results are special cases of ours.