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The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant. (English) Zbl 1113.11017

The aim of present paper is to study the generalized-Euler-constant function γ(z), defined for |z|1 by γ(z)= n=1 z n-1 (1 n-logn+1 n) and extended analytically to -[1,) by the integral representation γ(z)= 0 1 0 1 1-x (1-xyz)(-logxy)dxdy. This function has some very important constants in heart; γ(1)=γ=0·577 is the “Euler’s constant”, γ(-1)=log4 π is the “alternating Euler constant”, γ(1 2)=2log2 σ with

σ=123= n=1 n 1/2 n =1·661

is one of “Somos’s quadratic recurrence constant” [N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (published electronically), Sequence A112302 (2003; Zbl 1044.11108)].

The authors give various relations between this function and some well known special functions such as the Riemann zeta-function ζ(s)= n=1 n -s and the polylogarithm function Li k (z)= n=1 z n n -k , which lead to some interesting formulas and expansions for the above mentioned constants. They define

σ t =12 t3 t t= n=1 n 1/t n

for t>1 as a generalization of Somos’s quadratic recurrence constant, and they prove γ(1 t)=tlogt (t-1)σ t t-1 . They calculate γ(z) and γ ' (z) at roots of unity, which end in evaluating several related series and infinite products. Calculation of γ ' (z) is based on the Kinkelin-Bendersky hyperfactorial K function, the Weierstrass products for the gamma and Barnes G functions, and Jonquiere’s relation for the polylogarithm. These methods allow them to evaluate some double integrals, and especially considering the “Glaisher-Kinkelin constant” A=lim n 1 1 2 2 n n n -n 2 +n 2-1 12 e n 2 4 =1·282, they show that

γ ' (-1)=log2 11/6 A 6 π 3/2 eandγ '' (-1)=log2 10/3 A 24 π 4 e 13/4 -7ζ(3) 2π 2 ·

The paper contains some delicate computations and methods of analysis.

11B99Sequences and sets
33B99Elementary classical functions
11Y60Evaluation of constants