Asymptotic Formulas for Sequences Arising in the Problem of Approximating the Euler Number

We consider the function $$h(z)=\exp\{(1/z)\ln(1+z)\}=(1+z)^{1/z}$$ holomorphic in the complex plane with a cut along the ray $$(-\infty,-1]$$ of the real line. The coefficients of the power series expansion of the function $$h$$ , converging in the unit disk, form an alternating sequence of rational multiples of $$e$$ whose absolute values decrease to $$1$$ . Expanded asymptotic formulas are proved for both the coefficient sequence itself and the sequence of finite differences of an arbitrary fixed order associated with the sequence of absolute values of the coefficients. The study is motivated by various aspects of the problem of rational approximation of the number $$e$$ , as well as by one unsolved problem from the theory of finite differences.