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The generalized-Euler-constant function $\gamma \left(z\right)$ 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 $\gamma \left(z\right)$, defined for $|z|\le 1$ by $\gamma \left(z\right)={\sum }_{n=1}^{\infty }{z}^{n-1}\left(\frac{1}{n}-log\frac{n+1}{n}\right)$ and extended analytically to $ℂ-\left[1,\infty \right)$ by the integral representation $\gamma \left(z\right)={\int }_{0}^{1}{\int }_{0}^{1}\frac{1-x}{\left(1-xyz\right)\left(-logxy\right)}\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dy$. This function has some very important constants in heart; $\gamma \left(1\right)=\gamma =0·577\cdots$ is the “Euler’s constant”, $\gamma \left(-1\right)=log\frac{4}{\pi }$ is the “alternating Euler constant”, $\gamma \left(\frac{1}{2}\right)=2log\frac{2}{\sigma }$ with

$\sigma =\sqrt{1\sqrt{2\sqrt{3\cdots }}}=\prod _{n=1}^{\infty }{n}^{1/{2}^{n}}=1·661\cdots$

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 $\zeta \left(s\right)={\sum }_{n=1}^{\infty }{n}^{-s}$ and the polylogarithm function ${\text{Li}}_{k}\left(z\right)={\sum }_{n=1}^{\infty }{z}^{n}{n}^{-k}$, which lead to some interesting formulas and expansions for the above mentioned constants. They define

${\sigma }_{t}=\sqrt[t]{1\sqrt[t]{2}\sqrt[t]{3}\cdots }=\prod _{n=1}^{\infty }{n}^{1/{t}^{n}}$

for $t>1$ as a generalization of Somos’s quadratic recurrence constant, and they prove $\gamma \left(\frac{1}{t}\right)=tlog\frac{t}{\left(t-1\right){\sigma }_{t}^{t-1}}$. They calculate $\gamma \left(z\right)$ and ${\gamma }^{\text{'}}\left(z\right)$ at roots of unity, which end in evaluating several related series and infinite products. Calculation of ${\gamma }^{\text{'}}\left(z\right)$ 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” $A={lim}_{n\to \infty }{1}^{1}{2}^{2}\cdots {n}^{n}{n}^{-\frac{{n}^{2}+n}{2}-\frac{1}{12}}{e}^{\frac{{n}^{2}}{4}}=1·282\cdots$, they show that

${\gamma }^{\text{'}}\left(-1\right)=log\frac{{2}^{11/6}{A}^{6}}{{\pi }^{3/2}e}\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{\gamma }^{\text{'}\text{'}}\left(-1\right)=log\frac{{2}^{10/3}{A}^{24}}{{\pi }^{4}{e}^{13/4}}-\frac{7\zeta \left(3\right)}{2{\pi }^{2}}·$

The paper contains some delicate computations and methods of analysis.

##### MSC:
 11B99 Sequences and sets 33B99 Elementary classical functions 11Y60 Evaluation of constants