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

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.

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• A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind
• Generalizations of the Euler-Mascheroni constant associated with the hyperharmonic numbers
• Simulating a coin with irrational bias using rational arithmetic
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• Closed-form formula of Riemann zeta function and eta function for all non-zero given complex numbers via sums of powers of complex functions to disprove Riemann Hypothesis
• A note on some constants related to the zeta-function and their relationship with the Gregory coefficients
• Note on the Stieltjes constants: series with Stirling numbers of the first kind
• A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial