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This paper is devoted to Ser's and Hasse's series representations for thezeta-functions, as well as to several closely related results. The notesconcerning Ser's and Hasse's representations are given as theorems, while therelated expansions are given either as separate theorems or as formulae insidethe remarks and corollaries. In the first theorem, we show that the famousHasse's series for the zeta-function, obtained in 1930 and named after theGerman mathematician Helmut Hasse, is equivalent to an earlier expression givenby a little-known French mathematician Joseph Ser in 1926. In the secondtheorem, we derive a similar series representation for the zeta-functioninvolving the Cauchy numbers of the second kind. In the third theorem, with theaid of some special polynomials, we generalize the previous results to theHurwitz zeta-function. In the fourth theorem, we obtain a similar series withGregory's coefficients of higher order. In the fifth theorem, we extend theresults of the third theorem to a class of Dirichlet series. As a consequence,we obtain several globally convergent series for the zeta-functions. We alsoshow that Hasse's series may be obtained much more easily by using the theoryof finite differences, and we demonstrate that there exist numerous series ofthe same nature. In the sixth theorem, we show that Hasse's series is a simpleparticular case of a more general class of series involving the Stirlingnumbers of the first kind. All the expansions derived in the paper lead, inturn, to the series expansions for the Stieltjes constants, including newseries with rational terms for Euler's constant, for the logarithm of thegamma-function, for the digamma and trigamma functions. Finally, in theAppendix, we prove an interesting integral representation for the Bernoullipolynomials of the second kind, formerly known as the Fontana-Besselpolynomials.

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