Integrals of products of Bernoulli polynomials. (English) Zbl 1215.11014

Summary: Extending known results on integrals of products of two or three Bernoulli polynomials with limits of integration 0 and 1, we obtain identities for such integrals with limits of integration 0 and \(x\), for a variable \(x\).
Proposition 1. For all \(k,m\geq 0\) we have \[ \int_0^x B_k(t)B_m(t)\,dt=\frac{k!m!}{(k+m+1)!} \sum_{j=0}^k (-1)^j\binom{k+m+1}{k-j} \left(B_{k-j}(x)B_{m+j+1}(x)-B_{k-j}B_{m+j+1}\right). \]
As applications we obtain certain quadratic and cubic identities for Bernoulli polynomials.


11B68 Bernoulli and Euler numbers and polynomials
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