Three notes on Ser's and Hasse's representations for the zetafunctions
Abstract
This paper is devoted to Ser's and Hasse's series representations for the zetafunctions, 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 zetafunction, obtained in 1930 and named after the German mathematician Helmut Hasse, is equivalent to an earlier expression given by a littleknown French mathematician Joseph Ser in 1926. In the second theorem, we derive a similar series representation for the zetafunction 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 zetafunction. 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 zetafunctions. 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 gammafunction, 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 FontanaBessel polynomials.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 arXiv:
 arXiv:1606.02044
 Bibcode:
 2016arXiv160602044B
 Keywords:

 Mathematics  Number Theory
 EPrint:
 Integers (Electronic Journal of Combinatorial Number Theory), vol. 18A, art. A3, pp. 145, 2018