Expansions of generalized Euler's constants into the series of polynomials in $\pi^{-2}$ and into the formal enveloping series with rational coefficients only

TL;DR

This paper introduces two new series expansions for generalized Euler's constants, involving polynomials in 7 with rational coefficients and Bernoulli numbers, improving convergence and estimation accuracy.

Contribution

It presents novel series expansions for Euler's constants using Stirling and Bernoulli numbers, with improved convergence and estimation methods.

Findings

First series converges better than Euler's original series.

Second semi-convergent series is simple and involves Bernoulli numbers.

Derived more accurate estimations for generalized Euler's constants.

Abstract

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $\gamma_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $\pi^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. 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.