Inequalities and Bounds for the Incomplete Gamma Function

Inequalities involving the incomplete gamma function are established. They are obtained using logarithmic convexity of some function associated with the function in question. Lower and upper bounds for the incomplete gamma function are also derived. Bounds for the error function erf are also established.     

The Incomplete Gamma Function Expressed as a Sum of Macdonald Functions

The incomplete gamma function is rewritten as a finite sum of Macdonald functions (modified Bessel functions of the second kind), K_n+1/2(z), allowing a class of integrals appearing in atomic physics to be easily evaluated.     

Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument

Various approaches to the numerical representation of the incomplete Gamma function (m+1/2,z) for complex arguments z and non-negative small integer indices m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, classical numerical methods of sampling the basic integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in the z-plane, which stores a static matrix of higher derivatives. This is the obvious extension to a procedure that is in common use for real-valued z.     

关于伽玛函数的单调性质

伽玛函数的单调性质和对数完全单调性质被获得了.     

An Incomplete Function Space

In this paper we construct complete, regular convergence vector spaces E and F such that _c(E,F), the space of all continuous linear mappings from E to F, endowed with the continuous convergence structure, is not complete.     

Accurate Estimates of the Gamma Function Involving the PSI Function

The main result of this article is to establish new and accurate approximations of the gamma function in terms of the digamma function, that improve a result of Alzer and Batir [Appl. Math. Lett. 20(2007):778–781].     

A Generalization of the Euler Gamma Function

We define a generalized Euler gamma function Λ^β(z), where the product is taken over powers of integers rather than integers themselves. Studying the associated spectral functions and in particular the zeta function, we obtain the main properties of Λ^β(z) and its asymptotic expansion for large values of the argument.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 87–91, 2005Original Russian Text Copyright © by M. Spreafico     

On the Self-Decomposability of Euler's Gamma Function

Let be Euler's Gamma function. We prove that, for all 0, > 0, > 0, > 0, the function (( + iz)/() ^{i z)} , z R ¹, is a self-decomposable characteristic function from the Thorin class and derive its explicit canonical form. Similarly to [1], we also describe several classes of Lévy-type stochastic processes related to .     

利用Gamma函数求积分的几种形式

借助幂函数与对数函数的变量替换对Gamma函数从形式上加以推广,使Gamma函数中指数函数部分为指数函数与幂函数或对数函数与幂函数的复合函数时仍可求值,以扩大Gamma函数的使用范围.     

An Asymptotic Expansion of the Double Gamma Function

The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for log G(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Arg z|<π is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Arg z.     

有关Gamma函数的一个双不等式

本文将文献[1]中的双边不等式从自然数推广至实数,证明了下面不等式成立:(x/e)~x(2πx)~(1/2)(1 1/(12x))<Γ(x 1)<(x/e)~x(2πx)~(1/2)(1 1/(12x-0.5)),其中x≥1.     

Analytically Continued Hypergeometric Expression of the Incomplete Beta Function

The Incomplete Beta Function is rewritten as a Hypergeometric Function that is the analytic continuation of the conventional form, a generalization of the finite series, which simpifies the Stieltjes transform of powers of a monomial divided by powers of a binomial.     

关于伽马函数的不等式与渐近展开式

给出伽马函数的一个渐近展开式.基于获得的结果,我们建立了伽马函数的不等式.     

On Ramanujan's Double Inequality for the Gamma Function

Ramanujan presented (without proof) the following double inequalityfor the gamma function: .Recently, Karatsuba established that these inequalities holdfor x 1. We show that this can be slightly improved: the inequalitieshold for all x 0, even if we replace 1/100 by where f(x) = (1/)³[(x + 1)(e/x)^x]⁶–8x³–4x²–x.Moreover, and 1/30 are the best possible constant terms. 2000Mathematics Subject Classification 33B15 (primary); 26D15, 26D20(secondary).     

关于伽马函数的不等式与渐近展开式（英文）

给出伽马函数的一个渐近展开式.基于获得的结果,我们建立了伽马函数的不等式.     

A Power Mean Inequality for the Gamma Function

 A classical inequality for Euler’s gamma function states that

The Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function Γ_n, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γ_n function and their applications to summation of series and infinite products.This work was supported by NFS grant CCR-0204003.2000 Mathematics Subject Classification: Primary—33E20, 33F99, 11M35, 11B73     

Asymptotic Formulas and Inequalities for the Gamma Function in Terms of the Tri-Gamma Function

Results in Mathematics - In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial n! and the gamma function Γ in terms of the tri-gamma function...