Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions

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 xx 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 tt and ss and also αα and ββ is given. Application of the new method leads to the improvement of many known approximation formulas of the Stirling’s type.