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 gamma(z) = Sigma(infinity)(k=1) Z(k-1)(1/k - In k+1/k) by its partial sum gamma(N-1)(z) when 0 < z < 1. 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.