Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant

In this paper, we provide an estimate for approximating the generalized-Euler-constant function γ(z)=∑k=1∞zk−1(1k−lnk+1k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma (z)=\sum_{k=1}^{\infty }z ^{k-1} (\frac{1}{k}-\ln \frac{k+1}{k} )$\end{document} by its partial sum γN−1(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma _{N-1}(z)$\end{document} when 0<z<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< z<1$\end{document}. We obtain an asymptotic expansion for the generalized-Euler-constant function and show that the coefficients of the asymptotic expansion are explicitly expressed by the Eulerian fractions. Also, we find a recurrence relation for those coefficients. Using its relation with the generalized-Euler-constant function, we establish two inequalities for the generalized Somos’ quadratic recurrence constant. Moreover, two asymptotic expansions for the natural logarithm of the generalized Somos quadratic recurrence constant are presented.