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 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 = 1 2 ( t + s − 1 ) and β = 1 4 ( 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.