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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 \(| z| \leq 1\) by \(\gamma(z)=\sum_{n=1}^{\infty}z^{n-1}(\frac{1}{n}-\log\frac{n+1}{n})\) and extended analytically to \(\mathbb{C}-[1,\infty)\) 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” [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:
11B99 Sequences and sets
33B99 Elementary classical functions
11Y60 Evaluation of number-theoretic constants
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