Results 11 to 20 of about 607,360 (246)

Inequalities for the gamma and q-gamma functions

Journal of Approximation Theory, 2007
\textit{N. Batir} [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 4, Paper No. 97 (2004; Zbl 1078.33001)] found some symmetrical upper and lower bounds for \(\Gamma(x)\) in terms of \(\psi(x)\). In this study, the authors improved Batir's results by getting sharp inequalities.
Horst Alzer, Arcadii Z. Grinshpan
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Integral Representations of a Generalized Linear Hermite Functional

Mathematics, 2023
In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.
Roberto S. Costas-Santos
doaj   +1 more source

Multiple Gamma Functions and Multiple $q$-Gamma Functions

Publications of the Research Institute for Mathematical Sciences, 1997
We give an asymptotic expansion ( the higher Stirling formula ) and an infinite product representation ( the Weierstrass canonical product representation ) of the Vigneras multiple gamma functions by considering the classical limit of the multiple
Ueno, Kimio, Nishizawa, Michitomo
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On Complex Gamma-Function Integrals [PDF]

Symmetry, Integrability and Geometry: Methods and Applications, 2020
It was observed recently that relations between matrix elements of certain operators in the ${\rm SL}(2,\mathbb R)$ spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with ${\rm SL}(2,\mathbb C)$ symmetry group and ${\rm L}_2(\mathbb C)$ as a local Hilbert space give rise to a new type of $Γ ...
Derkachov, Sergey É., Manashov, Alexander N.   +1 more
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Monotonicity and inequalities for the gamma function

Journal of Inequalities and Applications, 2017
In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function x ↦ 1 24 x ( ln Γ ( x + 1 / 2 ) − x ln x + x − ln 2 π ) + 1 − 120 7 x 2 $$ x\mapsto \frac{1}{24x ( \ln \Gamma ( x+1/2 ) -x\ln x+x- \ln \sqrt{
Zhen-Hang Yang, Jing-Feng Tian
doaj   +1 more source

Completely monotonic functions involving the gamma and $q$-gamma functions [PDF]

Proceedings of the American Mathematical Society, 2005
The authors provide new examples of logarithmically completely monotonic functions. These examples are ratios of products involving the Gamma function or the \(q\)-Gamma function. As an application of these results new examples of infinitely divisible probability distributions can be obtained.
Grinshpan, Arcadii Z., Ismail, Mourad E.H.   +1 more
openaire   +2 more sources

Evaluation of the real parts of polylogarithm expressions containing complex arguments via certain logarithmic integrals [PDF]

Notes on Number Theory and Discrete Mathematics
We consider a polylogarithm expression containing complex arguments, namely 𝓟±(n)=ℜ(Liₙ((1±i)/2)). The central notion of the present paper is to evaluate the real parts of 𝓟±(n) for first four orders, specifically n = 1,2,3, and 4, by constructing ...
Narendra Bhandari
doaj   +1 more source

Recurrent identities for two special functions of hypergeometric type

Вестник Самарского университета: Естественнонаучная серия, 2023
The article presents conclusions and proofs of Gauss-type identities for two known hypergeometric type functions. For the derivation and justification of formulas, the representation of functions in the form of a series is used, as well as an integral ...
Svetlana V. Podkletnova
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On approximating the modified Bessel function of the second kind

Journal of Inequalities and Applications, 2017
In the article, we prove that the double inequalities π e − x 2 ( x + a ) < K 0 ( x ) < π e − x 2 ( x + b ) , 1 + 1 2 ( x + a ) < K 1 ( x ) K 0 ( x ) < 1 + 1 2 ( x + b ) $$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt ...
Zhen-Hang Yang, Yu-Ming Chu
doaj   +1 more source

The Multiple Gamma-Functions and the Log-Gamma Integrals [PDF]

International Journal of Mathematics and Mathematical Sciences, 2012
In this paper, which is a companion paper to [W], starting from the Euler integral which appears in a generalization of Jensen’s formula, we shall give a closed form for the integral of log . This enables us to locate the genesis of two new functions and considered by Srivastava and Choi. We consider the closely related functionA(a)and the Hurwitz zeta
X.-H. Wang, Y.-L. Lu
openaire   +3 more sources