Journal ArticleDOI
Integral representation of some functions related to the Gamma function
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In this paper, it was shown that the functions Θ(Θ(x) = [Gamma (x + 1)]^{1/x} (1 + 1/x)^x /x) and Θ (Θ (x, Θ) = Θ((Θ + 1)/x)) are Stieltjes transforms.Citations
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Proceedings ArticleDOI
Two Classes of Logarithmically Completely Monotonic Functions Associated with the Gamma Function
Journal ArticleDOI
Logarithmically complete monotonicity of a matrix-parametrized analogue of the multinomial distribution
TL;DR: In this article , a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions is introduced. But the authors show the logarithmic complete monotonicity of this generalization and derive new inequalities involving ratios of multivariate Gamma functions.
Journal ArticleDOI
Inequalities and complete monotonicity for the gamma and related functions
Chao-Ping Chen,Junesang Choi +1 more
Journal ArticleDOI
Logarithmically Complete Monotonicity of a Function Involving the Gamma Functions
TL;DR: In this paper, it was shown that a certain function involving ratio of the Euler gamma functions and some parameters is completely and logarithmically completely monotonic on the negative real axis, and that it can be represented by a Laplace-Stieitjes integral with non-decreasing determining function and converse.
Posted Content
A family of Horn-Bernstein functions
TL;DR: In this article, a family of recently investigated Bernstein functions is revisited and those functions for which the derivatives are logarithmically completely monotonic are identified, which leads to the definition of a class of Bernstein functions, which we propose to call Horn-Bernstein functions.
References
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Book
Table of Integrals, Series, and Products
TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
A table of integrals
TL;DR: Basic Forms x n dx = 1 n + 1 x n+1 (1) 1 x dx = ln |x| (2) udv = uv − vdu (3) 1 ax + bdx = 1 a ln|ax + b| (4) Integrals of Rational Functions