Neven Elezović

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.

Integral representations and integral transforms of some families of Mathieu type series

By using some integral representations for several Mathieu type series (see P.L. Butzer, T.K. Pogány, and H.M. Srivastava, A linear ODE for the Omega function associated with the Euler function E α(z) and the Bernoulli function B α(z), Appl. Math. Lett. 19 (2006), pp. 1073–1077; P. Cerone and C.T. Lenard, On integral forms of generalised Mathieu series, J. Inequal. Pure Appl. Math. 4 (5) (2003), Article 100, pp. 1–11 (electronic), T.K. Pogány; H.M. Srivastava and Ž. Tomovski, Some families of Mathieu a-series and alternating Mathieu a-series, Appl. Math. Comput. 173 (2006), pp. 69–108; H.M. Srivastava and Ž. Tomovski, Some problems and solutions involving Mathieus series and its generalizations, J. Inequal. Pure Appl. Math. 5 (2) (2004), Article 45, pp. 1–13 (electronic); Ž. Tomovski, Integral representations of generalized Mathieu series via Mittag-Leffler type functions, Fract. Calc. Appl. Anal. 10 (2007), pp. 127–138.) via the Bessel function J ν of the first kind, the Gauss hypergeometric function 2 F 1, the generalized hypergeometric function p F q and the Fox–Wright generalization p Ψ q of the hypergeometric function p F q , a number of integral representations of the Laplace, Fourier, and Mellin types are derived here for certain general families of Mathieu type series. Some interesting corollaries and consequences of these integral representations are also considered.

On some inequalities for beta and gamma functions via some classical inequalities.

We improve several results recently established by Dragomir et al. in (2000) for the Gamma and Beta functions. All we need is some clever applications of classical inequalities.

Asymptotic expansions of integral means and applications to the ratio of gamma functions

Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form B(A(x))=C(x), where asymptotic expansions of B and C are known. The results are illustrated by calculation of some important integral means connected with gamma function.

Convexity andq-gamma function

Inequalities and convexity properties known for the gamma function are extended to theq-gamma function, 0<q<1. Applying some classical inequalities for convex functions, we deduce monotonicity results for several functions involving theq-gamma function. Further applications to the probability theory are given.

Estimations of psi function and harmonic numbers

The asymptotic expansion of digamma function is a starting point for the derivation of approximants for harmonic sums or Euler-Mascheroni constant. It is usual to derive such approximations as values of logarithmic function, which leads to the expansion of the exponentials of digamma function. In this paper the asymptotic expansion of the function exp ( p ? ( x + t ) ) is derived and analyzed in details, especially for integer values of parameter p. The behavior for integer values of p is proved and as a consequence a new identity for Bernoulli polynomials. The obtained formulas are used to improve know inequalities for Eulers constant and harmonic numbers.

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.

Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions

We present new asymptotic expansions of the logarithm of the gamma function in terms of the psi and polygamma functions. Based on these expansions, we prove new complete monotonicity properties of some functions involving the gamma, psi and polygamma functions.