Arithmetical properties of some series with logarithmic coefficients

Abstract

We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler's constant γ , log 4 π and log σ , where σ is Somos's quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.

Asymptotic expansions for certain mathematical constants and special functions

“…The constant σ appears in important problems from pure and applied analysis, which is a motivation of a large number of papers (see, e.g., [12,14,16,33,35,39,40,41,43,47,48,52]).…”

Bounds for the generalized Euler-constant function

“…Simultaneously, Pilehrood and Pilehrood [2] studied it and showed that zγ (z) is continuous on the closed unit disc D = {z ∈ C : |z| ≤ 1}, holomorphic in its interior. It is clear from the definition of γ (z) that γ (1) = γ .…”

“…Also, it is known that γ (−1) = log 4 π (see [1]). γ (−1) is known to be 'alternating Euler-constant' and it is closely related with some other mathematical constants such as Somos' quadratic recurrence constant σ (see [1][2][3][4][5][6][7] Classroom Note since γ (1/2) = 2 log 2 σ . The constant σ arose when Somos [7] had examined the asymptotic behaviour of the sequence (g n ) defined by g 0 = 1, and g n = ng 2 n−1 , n ≥ 1.…”

Inequalities and asymptotic expansions related to the generalized Somos quadratic recurrence constant

“…Pilehrood and Pilehrood [13] considered the function (). The function generalizes both Euler’s constant and the alternating Euler constant [17, 18].…”