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Zbl 0552.10007
Todorov, Pavel G.
On the theory of the Bernoulli polynomials and numbers.
(English)
[J] J. Math. Anal. Appl. 104, 309-350 (1984). ISSN 0022-247X

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 $B\sb n$, generated by the Taylor expansion $g(t)=t/(e\sp t- 1)=\sum\sp{\infty}\sb{n=0}B\sb nt\sp n/n!\quad (\vert t\vert <2\pi),$ are represented by the classical formula $B\sb n=\sum\sp{n}\sb{k=0}(-1)\sp k k! S(n,k)/(k+1).$ The author first obtains an explicit formula for the nth derivative of g(t). Namely, $g\sp{(n)}(t)=\sum\sp{n}\sb{k=0}(-1)\sp k k! S(n,k) G\sb k(t)$ in the finite t-plane punctured at the points 2m$\pi$ i, $m=\pm 1,\pm 2,...$, where the functions $G\sb k(t)$ are regular for the considered t and have the representation $$G\sb 0(t)=g(t),\quad G\sb k(t)=e\sp{-t}/(1-e\sp{-t})\sp{k+1}[t- \sum\sp{k}\sb{\nu =1}(1-e\sp{-t})\sp{\nu}/\nu]\quad (1\le k\le n;\quad n\ge 1).$$ He then obtains the nth derivative of the generating function $g(t,x)=te\sp{tx}/(e\sp t-1)=\sum\sp{\infty}\sb{n=0}B\sb n(x)t\sp n/n!\quad (\vert t\vert <2\pi)$ of the Bernoulli polynomials $B\sb n(x)=\sum\sp{n}\sb{\nu =0}\left( \matrix n\\ \nu \endmatrix \right)B\sb{\nu}x\sp{n-\nu}.$ Namely, $\partial\sp ng(t,x)/\partial t\sp n=e\sp{tx}\sum\sp{n}\sb{k=0}(-1)\sp k \Delta\sp kx\sp n G\sb k(t)$ in the finite t-plane punctured at the points 2m$\pi$ i, $m=\pm 1,\pm 2,...$, where $\Delta\sp kx\sp n$ is the finite difference of the kth order of $x\sp n$. In particular, for $t=0$, he finds the new formula $B\sb n(x)=\sum\sp{n}\sb{k=0}(-1)\sp k \Delta\sp kx\sp n/(k+1),$ which generalizes the classical representation of $B\sb n$ given above. He also derives a generalization of the Kronecker-Bergmann formula for $B\sb n$ to $B\sb n(x).$ \par The author then proceeds to introduce the class of rational functions $T\sb n(z)=\sum\sp{\infty}\sb{k=0}(-1)\sp kS(n,k)/(z+k)$ and to establish an analytic expression of $T\sb n(z)$ by means of any of the functions $T\sb{n-\nu}$ $(\nu =0,1,...,n)$. As a corollary, he derives a series of new representations of the Bernoulli numbers $B\sb n=T\sb n(1)$. Among other results, he also obtains a representation of $T\sb n(z)$ as a quotient of two relatively prime polynomials.
[A.N.Philippou]
MSC 2000:
*11B39 Special numbers, etc.
05A15 Combinatorial enumeration problems
05A19 Combinatorial identities

Keywords: new representations; Bernoulli polynomials; Bernoulli numbers; Taylor expansion; nth derivative; nth derivative of the generating function; generalization of the Kronecker-Bergmann formula; rational functions

Cited in: Zbl 1171.11010

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