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

Abstract

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.

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]

“…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…”

“…186-187], [159,vol. II,, [10,32,33,34,21,62] [152,61,189,126,14,188,174,80,27,26,83,4,175,70,117,163,164,154,148,149,76,105,18]. Note that many writers discovered these numbers independently, without realizing that they deal with the 13 In the explicit form, this integral formula was given by Franel in 1895 [56] (in the above, we corrected the original Franel's formula which was not valid for m = 0).…”

“…and hence, may be neglected at large n. 18 Note that for fixed n, the left-hand side of (26) reaches its maximum when z is imaginary pure. the joint analysis of Figs.…”

“…Higher generalized Euler's constants are not known to be related to the "standard" mathematical constants, nor to the "classic" functions of analysis. In our recent work [18], we obtained two interesting series representations for the logarithm of the Γ-function containing Stirling numbers of the first kind S 1 (n, k)…”

“…Further generalization of both, generalized Euler's constants defined accordingly to (2) and generalized Euler's constants introduced by Briggs and Lehmer, was done by Dilcher in [49]. 4 Both series converge in a part of the right half-plane [18,Fig. 2] at the same rate as ∑ n ln m n −2 , where m = 1 for ln Γ(z) and Ψ(z), m = 2 for Ψ 1 (z) and Ψ 2 (z), m = 3 for Ψ 3 (z) and Ψ 4 (z), etc.…”