Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions

✍ Scribed by Tomislav Burić; Neven Elezović

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 281 KB
• Volume: 235
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2011.01.045

✦ Synopsis

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

, 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 + ts)(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.