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The generalized-Euler-constant function $\gamma (z)$ 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 $\gamma(z)$, defined for $\vert z\vert \leq 1$ by $\gamma(z)=\sum_{n=1}^{\infty}z^{n-1}(\frac{1}{n}-\log\frac{n+1}{n})$ and extended analytically to $\Bbb{C}-[1,\infty)$ by the integral representation $\gamma(z)=\int_0^1 \int_0^1\frac{1-x}{(1-xyz)(-\log xy)}\,dx\,dy$. This function has some very important constants in heart; $\gamma(1)=\gamma=0.577\cdots$ is the “Euler’s constant”, $\gamma(-1)=\log\frac{4}{\pi}$ is the “alternating Euler constant”, $\gamma(\frac{1}{2})=2\log\frac{2}{\sigma}$ 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” [{\it 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 $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ and the polylogarithm function $\text{Li}_k(z)=\sum_{n=1}^{\infty}z^n n^{-k}$, which lead to some interesting formulas and expansions for the above mentioned constants. They define $$\sigma_t=\root t\of {1\root t\of 2{\root t\of 3\cdots}}=\prod_{n=1}^{\infty}n^{1/t^n}$$ for $t>1$ as a generalization of Somos’s quadratic recurrence constant, and they prove $\gamma(\frac{1}{t})=t\log\frac{t}{(t-1)\sigma_t^{t-1}}$. They calculate $\gamma(z)$ and $\gamma'(z)$ at roots of unity, which end in evaluating several related series and infinite products. Calculation of $\gamma'(z)$ 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” $A=\lim_{n\rightarrow\infty}1^12^2\cdots n^n n^{-\frac{n^2+n}{2}-\frac{1}{12}}e^{\frac{n^2}{4}}=1.282\cdots$, they show that $$\gamma'(-1)=\log\frac{2^{11/6}A^6}{\pi^{3/2}e} \quad\text{and}\quad \gamma''(-1)=\log\frac{2^{10/3}A^{24}}{\pi^4e^{13/4}}-\frac{7\zeta(3)}{2\pi^2}.$$ The paper contains some delicate computations and methods of analysis.

MSC:
11B99Sequences and sets
33B99Elementary classical functions
11Y60Evaluation of constants
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References:
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