Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions

摘要

The main subject of this paper is the analysis of asymptotic expansions of Wallis quotient function Γ(x+t)Γ(x+s) and Wallis power function [Γ(x+t)Γ(x+s)]1/(t−s), when x tends to infinity. Coefficients of these expansions are polynomials derived from Bernoulli polynomials. The key to our approach is the introduction of two intrinsic variables α=12(t+s−1) and β=14(1+t−s)(1−t+s) which are naturally connected with Bernoulli polynomials and Wallis functions. Asymptotic expansion of Wallis functions in terms of variables t and s and also α and β is given. Application of the new method leads to the improvement of many known approximation formulas of the Stirling’s type.