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On the theory of the Bernoulli polynomials and numbers. (English) Zbl 0552.10007

This is an excellent paper containing several new representations of the Bernoulli polynomials and the Bernoulli numbers. In the sequel, let n be any nonnegative integer unless otherwise specified, and let S(n,k) be the Stirling numbers of the second kind. It is well known that the Bernoulli numbers ${B}_{n}$, generated by the Taylor expansion $g\left(t\right)=t/\left({e}^{t}-1\right)={\sum }_{n=0}^{\infty }{B}_{n}{t}^{n}/n!\phantom{\rule{1.em}{0ex}}\left(|t|<2\pi \right),$ are represented by the classical formula ${B}_{n}={\sum }_{k=0}^{n}{\left(-1\right)}^{k}k!S\left(n,k\right)/\left(k+1\right)·$ The author first obtains an explicit formula for the nth derivative of g(t). Namely, ${g}^{\left(n\right)}\left(t\right)={\sum }_{k=0}^{n}{\left(-1\right)}^{k}k!S\left(n,k\right){G}_{k}\left(t\right)$ in the finite t-plane punctured at the points 2m$\pi$ i, $m=±1,±2,···$, where the functions ${G}_{k}\left(t\right)$ are regular for the considered t and have the representation

${G}_{0}\left(t\right)=g\left(t\right),\phantom{\rule{1.em}{0ex}}{G}_{k}\left(t\right)={e}^{-t}/{\left(1-{e}^{-t}\right)}^{k+1}\left[t-\sum _{\nu =1}^{k}{\left(1-{e}^{-t}\right)}^{\nu }/\nu \right]\phantom{\rule{1.em}{0ex}}\left(1\le k\le n;\phantom{\rule{1.em}{0ex}}n\ge 1\right)·$

He then obtains the nth derivative of the generating function $g\left(t,x\right)=t{e}^{tx}/\left({e}^{t}-1\right)={\sum }_{n=0}^{\infty }{B}_{n}\left(x\right){t}^{n}/n!\phantom{\rule{1.em}{0ex}}\left(|t|<2\pi \right)$ of the Bernoulli polynomials ${B}_{n}\left(x\right)={\sum }_{\nu =0}^{n}\left(\begin{array}{c}n\\ \nu \end{array}\right){B}_{\nu }{x}^{n-\nu }·$ Namely, ${\partial }^{n}g\left(t,x\right)/\partial {t}^{n}={e}^{tx}{\sum }_{k=0}^{n}{\left(-1\right)}^{k}{{\Delta }}^{k}{x}^{n}{G}_{k}\left(t\right)$ in the finite t-plane punctured at the points 2m$\pi$ i, $m=±1,±2,···$, where ${{\Delta }}^{k}{x}^{n}$ is the finite difference of the kth order of ${x}^{n}$. In particular, for $t=0$, he finds the new formula ${B}_{n}\left(x\right)={\sum }_{k=0}^{n}{\left(-1\right)}^{k}{{\Delta }}^{k}{x}^{n}/\left(k+1\right),$ which generalizes the classical representation of ${B}_{n}$ given above. He also derives a generalization of the Kronecker-Bergmann formula for ${B}_{n}$ to ${B}_{n}\left(x\right)·$

The author then proceeds to introduce the class of rational functions ${T}_{n}\left(z\right)={\sum }_{k=0}^{\infty }{\left(-1\right)}^{k}S\left(n,k\right)/\left(z+k\right)$ and to establish an analytic expression of ${T}_{n}\left(z\right)$ by means of any of the functions ${T}_{n-\nu }$ $\left(\nu =0,1,···,n\right)$. As a corollary, he derives a series of new representations of the Bernoulli numbers ${B}_{n}={T}_{n}\left(1\right)$. Among other results, he also obtains a representation of ${T}_{n}\left(z\right)$ as a quotient of two relatively prime polynomials.

Reviewer: A.N.Philippou

##### MSC:
 11B39 Fibonacci and Lucas numbers, etc. 05A15 Exact enumeration problems, generating functions 05A19 Combinatorial identities, bijective combinatorics