On the theory of the Bernoulli polynomials and numbers
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 \(g(t)=t/(e^ t- 1)=\sum^{\infty}_{n=0}B_ nt^ n/n!\quad (| t| <2\pi),\) are represented by the classical formula \(B_ n=\sum^{n}_{k=0}(-1)^ k k! S(n,k)/(k+1).\) The author first obtains an explicit formula for the nth derivative of g(t). Namely, \(g^{(n)}(t)=\sum^{n}_{k=0}(-1)^ k k! S(n,k) G_ k(t)\) in the finite t-plane punctured at the points 2m\(\pi\) i, \(m=\pm 1,\pm 2,...\), where the functions \(G_ k(t)\) are regular for the considered t and have the representation \[ G_ 0(t)=g(t),\quad G_ k(t)=e^{-t}/(1-e^{-t})^{k+1}[t- \sum^{k}_{\nu =1}(1-e^{-t})^{\nu}/\nu]\quad (1\leq k\leq n;\quad n\geq 1). \] He then obtains the nth derivative of the generating function \(g(t,x)=te^{tx}/(e^ t-1)=\sum^{\infty}_{n=0}B_ n(x)t^ n/n!\quad (| t| <2\pi)\) of the Bernoulli polynomials \(B_ n(x)=\sum^{n}_{\nu =0}\left( \begin{matrix} n\\ \nu \end{matrix} \right)B_{\nu}x^{n-\nu}.\) Namely, \(\partial^ ng(t,x)/\partial t^ n=e^{tx}\sum^{n}_{k=0}(-1)^ k \Delta^ kx^ n G_ k(t)\) in the finite t-plane punctured at the points 2m\(\pi\) i, \(m=\pm 1,\pm 2,...\), where \(\Delta^ kx^ n\) is the finite difference of the kth order of \(x^ n\). In particular, for \(t=0\), he finds the new formula \(B_ n(x)=\sum^{n}_{k=0}(-1)^ k \Delta^ kx^ n/(k+1),\) which generalizes the classical representation of \(B_ n\) given above. He also derives a generalization of the Kronecker-Bergmann formula for \(B_ n\) to \(B_ n(x).\) The author then proceeds to introduce the class of rational functions \(T_ n(z)=\sum^{\infty}_{k=0}(-1)^ kS(n,k)/(z+k)\) and to establish an analytic expression of \(T_ n(z)\) by means of any of the functions \(T_{n-\nu}\) \((\nu =0,1,...,n)\). As a corollary, he derives a series of new representations of the Bernoulli numbers \(B_ n=T_ n(1)\). Among other results, he also obtains a representation of \(T_ n(z)\) as a quotient of two relatively prime polynomials.
nth derivative of the generating function
generalization of the Kronecker-Bergmann formula
scientific article; zbMATH DE number 3879006