Explicit formulas for Bernoulli polynomials of order \(n\)
The author shows that the \(n\)-th order Hurwitz zeta function \(\zeta_n (s,x)\), defined as the meromorphic continuation of the series \(\sum (x+k_1+ \cdots+ k_n)^{-s}\) extended over all nonnegative integers \(k_1, \dots, k_n\), can be expressed in terms of classic (first order) Hurwitz zeta functions as a linear combination of the form \[ \zeta_n(s,x) = \sum^{n-1}_{j=0}p_{n,j} (x)\zeta (s-j,x), \] where the multipliers are polynomials in \(x\) whose coefficients involve Stirling numbers of the first kind. By calculating the residue at \(s=k\) of each member of the last equation, the author obtains an explicit formula expressing Bernoulli polynomials of order \(n\) as a finite double sum involving classic Bernoulli polynomials (of order 1) and Stirling numbers.
scientific article; zbMATH DE number 941303