Summability Calculus

Journal of Mathematical Analysis and Applications

2016

In this paper, two new series for the logarithm of the Γ-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to π −1 . In particular, for any value of the form ln Γ( 1 2 n ± απ −1 ) and Ψ k ( 1 2 n ± απ −1 ), where Ψ k stands for the kth polygamma function, α is positive rational greater than 1 6 π, n is integer and k is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series converge uniformly at the same rate as ∑(n ln m n) −2 , where m = 1, 2, 3, . . . , depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the most attracting values, such as ln. Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gregory's coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gregory's coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. In particular, these include sharp bounds for Gregory's coefficients and for the Cauchy numbers of the second kind.

Section: I1 Motivation Of the Studymentioning

confidence: 99%

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

Journal of Mathematical Analysis and Applications

2016

“…which follows from the Stirling formula for the Γ-function. 9 Unsigned (or signless) and signed Stirling numbers of the first kind, which are also known as factorial coefficients, are denoted as |S 1 (n, l)| and S 1 (n, l) respectively (the latter are related to the former as S 1 (n, l) = (−1) n±l |S 1 (n, l)|). 10 Because in literature various names, notations and definitions were adopted for the Stirling numbers of the first kind, we specify that we use exactly the same definitions and notation as in [18,Section 2.1], that is to say |S 1 (n, l)| and S 1 (n, l) are defined as the coefficients in the expansion of rising/falling factorial…”

Section: I2 Notations and Some Definitionsmentioning

confidence: 99%

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

Journal of Number Theory

2016

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.

“…9 Unsigned (or signless) and signed Stirling numbers of the first kind, which are also known as factorial coefficients, are denoted as |S 1 (n, l)| and S 1 (n, l) respectively (the latter are related to the former as S 1 (n, l) = (−1) n±l |S 1 (n, l)|). 10 Because in literature various names, notations and definitions were adopted for the Stirling numbers of the first kind, we specify that we use exactly the same definitions and notation as in [18,Section 2.1], that is to say |S 1 (n, l)| and S 1 (n, l) are defined as the coefficients in the expansion of rising/falling factorial…”

Section: I2 Notations and Some Definitionsmentioning

confidence: 99%

“…Other notations are standard. 9 A simpler variant of the above formula may be found in [177]. 10 There exist more than 50 notations for the Stirling numbers, see e.g.…”

Section: I2 Notations and Some Definitionsmentioning

confidence: 99%

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

Journal of Number Theory

2017

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.