Three notes on Ser's and Hasse's representations for the zeta-functions

TL;DR

This paper explores Ser's and Hasse's series representations for zeta-functions, establishing equivalences, generalizations, and new series expansions, including for the Hurwitz zeta-function and related constants, using various mathematical tools.

Contribution

It introduces new series representations for zeta-functions, generalizes existing results, and connects these series to broader mathematical concepts like Stirling numbers and Bernoulli polynomials.

Findings

Hasse's series is equivalent to Ser's series from 1926.

Derived new series involving Cauchy numbers and Gregory's coefficients.

Extended results to Hurwitz zeta and Dirichlet series, providing new convergent series.

Abstract

This paper is devoted to Ser's and Hasse's series representations for the zeta-functions, as well as to several closely related results. The notes concerning Ser's and Hasse's representations are given as theorems, while the related expansions are given either as separate theorems or as formulae inside the remarks and corollaries. In the first theorem, we show that the famous Hasse's series for the zeta-function, obtained in 1930 and named after the German mathematician Helmut Hasse, is equivalent to an earlier expression given by a little-known French mathematician Joseph Ser in 1926. In the second theorem, we derive a similar series representation for the zeta-function involving the Cauchy numbers of the second kind. In the third theorem, with the aid of some special polynomials, we generalize the previous results to the Hurwitz zeta-function. In the fourth theorem, we obtain a similar series with Gregory's coefficients of higher order. In the fifth theorem, we extend the results of the third theorem to a class of Dirichlet series. As a consequence, we obtain several globally convergent series for the zeta-functions. We also show that Hasse's series may be obtained much more easily by using the theory of finite differences, and we demonstrate that there exist numerous series of the same nature. In the sixth theorem, we show that Hasse's series is a simple particular case of a more general class of series involving the Stirling numbers of the first kind. All the expansions derived in the paper lead, in turn, to the series expansions for the Stieltjes constants, including new series with rational terms for Euler's constant, for the logarithm of the gamma-function, for the digamma and trigamma functions. Finally, in the Appendix, we prove an interesting integral representation for the Bernoulli polynomials of the second kind, formerly known as the Fontana-Bessel polynomials.