Expansions of generalized Euler's constants into the series of polynomials inπ−2and into the formal enveloping series with rational coefficients only

Abstract

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γ m are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π −2 with rational coefficients and converges slightly better than Euler's series ∑ n −2. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions

“…By (18) it follows that the sequence n → f p n [g](x) converges. Denoting the limiting function by f, by (17) and assertion (a) we must have ∆f = g. Moreover, we also have f(½) = ¼ by Theorem 3.1.…”

“…We start with a technical but fundamental lemma. Recall first that, for any n ∈ N, the nth Gregory coefficient (also called the nth Bernoulli number of the second kind) is the number G n defined by the equation (see, e.g., [17][18][19]61])…”

“…and the latter series is called Fontana-Mascheroni's series (see, e.g., [17]). Thus, the series representation of the asymptotic constant σ[g] given in (42) provides the analogue of Fontana-Mascheroni's series for any function g satisfying the assumptions of Proposition 6.8.…”

“…Recall also that the numbers γ n = γ n (½), where n ∈ N, are called the Stieltjes constants. For recent background on these constants, see, e.g., Blagouchine [16,17] and Blagouchine and Coppo [19] (see also Nan-Yue and Williams [68]). These constants are known to satisfy γ ¼ (x) = −ψ(x) and γ ¼ = γ as well as the following identities for every q ∈ N…”

Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function

“…There is comprehensive literature on deriving series and integral representations for the Stieltjes constants and their extensions (see for example [4,5,13,14,15,33,22]). These representations usually allow a more accurate estimation of mentioned constants (see for example [1,3,5]).…”