Mező, István A new formula for the Bernoulli polynomials. (English) Zbl 1237.11010 Result. Math. 58, No. 3-4, 329-335 (2010). 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 \[ H(n,r)=\sum_{1\leq n_0+n_1+\dots+n_r\leq n}\frac{1}{n_0n_1\cdots n_r}\;. \] Reviewer: Roland Bacher (St. Martin d’Hères) Cited in 59 Documents MSC: 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers Keywords:Stirling numbers; \(r\)-Stirling numbers; Whitney numbers; Bernoulli polynomials; harmonic numbers; Stirling-type pairs; hyperharmonic numbers; harmonic polynomials PDFBibTeX XMLCite \textit{I. Mező}, Result. Math. 58, No. 3--4, 329--335 (2010; Zbl 1237.11010) Full Text: DOI References: [1] Benjamin A.T., Gaebler D., Gaebler R.: A combinatorial approach to hyperharmonic numbers. INTEGERS Electron J Combin Number Theory 3, 1–9 (2003) #A15 · Zbl 1128.11309 [2] Benoumhani M.: On Whitney numbers of Dowling lattices. Discrete Math. 159, 13–33 (1996) · Zbl 0861.05004 · doi:10.1016/0012-365X(95)00095-E [3] Broder A.Z.: The r-Stirling numbers. Discrete Math. 49, 241–259 (1984) · Zbl 0535.05006 · doi:10.1016/0012-365X(84)90161-4 [4] Cheon G.-S., El-Mikkawy M.E.A.: Generalized harmonic numbers with Riordan arrays. J. Number Theory 128(2), 413–425 (2008) · Zbl 1131.05011 · doi:10.1016/j.jnt.2007.08.011 [5] Dil A., Mezo I.: A symmetric algorithm for hyperharmonic and Fibonacci numbers. Appl. Math. Comput. 206, 942–951 (2008) · Zbl 1200.65104 [6] Hsu L.C., Shiue P.J.-S.: A unified approach to generalized Stirling numbers. Adv. Appl. Math. 20, 366–384 (1998) · Zbl 0913.05006 · doi:10.1006/aama.1998.0586 [7] Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics. Addison- Wesley, Reading (1989) · Zbl 0668.00003 [8] Mezo I., Dil A.: Euler–Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence. Cent. Eur. J. Math. 7(2), 310–321 (2009) · Zbl 1229.11043 · doi:10.2478/s11533-009-0008-5 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.