Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593548 | Journal of Number Theory | 2016 | 32 Pages |
In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γmγm are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π−2π−2 with rational coefficients and converges slightly better than Euler's series ∑n−2∑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.