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 ∞ z k − 1 ( 1 k − ln k + 1 k ) by its partial sum γ 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.