A note on explicit formulas for Bernoulli polynomials. (English) Zbl 1535.11031
Summary: For \(r\in\left \{1,-1,\frac{1}{2}\right\} \), we prove several explicit formulas for the \(n\)-th Bernoulli polynomial \(B_n\left(x \right)\), in which \(B_n\left(x\right)\) is equal to a linear combination of the polynomials \(x^n, \left(x+r\right)^n,\ldots,\left(x+rm\right)^n\), where \(m\) is any fixed positive integer greater than or equal to \(n\).
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
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