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Some results on sums of products of Bernoulli polynomials and Euler polynomials. (English) Zbl 1277.05007

Summary: 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 \[ S_n^{(k)}(y;l,k-l):= \sum _{\substack{ j_1+\cdots+j_k=n \\ j_1,\dots,j_k\geq 0 }} \binom {n}{j_1,\dots,j_k} B_{j_1}(x_1)\cdots B_{j_l}(x_l)E_{j_{l+1}}(x_{l+1}) \cdots E_{j_k}(x_k). \] 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.

MSC:

05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
11B68 Bernoulli and Euler numbers and polynomials
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