Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions

• Published in: Journal of computational and applied mathematics Vol. 235; no. 11; pp. 3315 - 3331
• Main Authors: BURIC, Tomislav, ELEZOVIC, Neven
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.04.2011; Elsevier
• Subjects: Algebra; Approximation; Asymptotic expansion; Asymptotic expansions; Bernoulli polynomials; Computation; Exact sciences and technology; Gamma function; Infinity; Mathematical analysis; Mathematical models; Mathematics; Number theory; Numerical analysis; Numerical analysis. Scientific computation; Quotients; Sciences and techniques of general use; Special functions; Stirling formula; Wallis power function; Wallis quotient; secondary; Wallis quotient; Asymptotic expansion; Stirling formula; Bernoulli polynomials; Gamma function; Wallis power function; primary; Polynomial; primary 33B15; Approximation; 41A60; 57Q55; 11B68; 33F05; Polynomial function; secondary 40A25; Numerical analysis; Applied mathematics
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2011.01.045

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.