Tomislav Burić

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 @C(x+t)@C(x+s) and Wallis power function [@C(x+t)@C(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 @a=12(t+s-1) and @b=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 t and s and also @a and @b is given. Application of the new method leads to the improvement of many known approximation formulas of the Stirlings type.

New asymptotic expansions of the quotient of gamma functions

Asymptotic expansions of the function [Γ(x+t)/Γ(x+s)]1/(t−s) involving exponential function are given and analysed. An efficient algorithm for calculating coefficients of these expansions is obtained. An application to the asymptotic expansion of the central binomial coefficient is given.

Approximants of the Euler-Mascheroni constant and harmonic numbers

In this paper various approximations for the Euler-Mascheroni constant and harmonic numbers are studied and improved and general asymptotic expansions related to this approximations are obtained.

Asymptotic expansions of the gamma function and Wallis function through polygamma functions

A new view on classical asymptotic expansions of the logarithm of gamma function is given. Then general formulae for the asymptotic expansions of the logarithm of gamma function and the Wallis power function through polygamma functions are derived and analysed.