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Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions

The main subject of this paper is analysis of asymptotic expansions of Wallis quotient function $\dfrac{; ; ; \Gamma(x+t)}; ; ; {; ; ; \Gamma(x+s)}; ; ; $ and Wallis power function $[\frac{; ; ; \Gamma(x+t)}; ; ; {; ; ; \Gamma(x+s)}; ; ; ]^{; ; ; 1/(t-s)}; ; ; $, when $x$ tends to infinity. Coefficients of these expansions are polynomials derived from Bernoulli polynomials. The key of our approach is introduction of two intrinsic variables $\alpha=\frac12(t+s-1)$ and $\beta=\tfrac14(1+t-s)(1-t+s)$ which are naturally connected with Bernoulli polynomials and Wallis functions. Asymptotic expansion of Wallis functions in terms of variables $t$ and $s$ and also $\alpha$ and $\beta$ is given. Application of the new method leads to the improvement of many known approximation formulas of the Stirling's type.

Gamma function; Wallis quotient; Wallis power function; Bernoulli polynomials; Asymptotic expansion; Stirling formula; binomial coefficient

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