The aim of present paper is to study the generalized-Euler-constant function , defined for by and extended analytically to by the integral representation . This function has some very important constants in heart; is the “Euler’s constant”, is the “alternating Euler constant”, with
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 and the polylogarithm function , which lead to some interesting formulas and expansions for the above mentioned constants. They define
for as a generalization of Somos’s quadratic recurrence constant, and they prove . They calculate and at roots of unity, which end in evaluating several related series and infinite products. Calculation of is based on the Kinkelin-Bendersky hyperfactorial function, the Weierstrass products for the gamma and Barnes functions, and Jonquiere’s relation for the polylogarithm. These methods allow them to evaluate some double integrals, and especially considering the “Glaisher-Kinkelin constant” , they show that
The paper contains some delicate computations and methods of analysis.