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A new formula for the Bernoulli polynomials. (English) Zbl 1237.11010

The author defines \(r\)-Whitney numbers \(w_{m,r}(n,k)\) and \(W_{m,r}(n,k)\) of the first and second kind by the equalities \[ m^nx^{\underline{n}}=\sum_{k=0}^nw_{m,r}(n,k)(mx+r)^k \] and \[ (mx+r)^n=\sum_{k=0}^nm^kW_{m,r}(n,k)x^{\underline{k}} \] with \(x^{\underline{n}}=x(x-1)\cdots(x-n+1)\) denoting falling factorials. These numbers are also given by \[ w_{m,r}(n,k)=\sum_{i=0}^n{n\choose i}m^{i-k}r^{n-i}S^2(i,k) \] (\(S^2(i,k)\) is probably a misprint and should be replaced by the Stirling number of the first kind \(S^1(i,k)\)) and \[ W_{m,r}(n,k)=\frac{1}{m^kk!}\sum_{i=0}^k{k\choose i}(-1)^{k-i}(mi+r)^n \] and specialize to Stirling numbers of the first and second kind for \(m=1\) and \(r=0\). They are a new example of a Stirling pair, a sequence of pairs of numbers satisfying equations also fulfilled by Stirling numbers of the first and second kind.
The main result of the paper is the identity \[ {n+1\choose l} B_{n-l+1}\left(\frac{r}{m}\right) =(n+1)\sum_{k=0}^n m^{k-n} W_{m,r}(n,k) \frac{S^1(k+1,l)}{k+1} \] (the formula in the paper has a misprint in the exponent of \(m\)) expressing the value of a Bernoulli polynomial at rational numbers in terms of the numbers \(W_{m,r}(n,k)\) and of Stirling numbers of the first kind.
The author also obtains the formula \[ H_{n-1}\left(1-\frac{r}{m}\right)=\frac{1}{m^{n-1}n!}| w_{m,r}(n,1)| \] for the value of the so-called harmonic polynomial \[ H_n(z)=\sum_{k=0}^n\frac{(-1)^kH(n+1,k)}{k!}z^k \] defined in terms of the generalized harmonic numbers \[ H(n,r)=\sum_{1\leq n_0+n_1+\dots+n_r\leq n}\frac{1}{n_0n_1\cdots n_r}\;. \]

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
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