Journal ArticleDOI
Integral representation of some functions related to the Gamma function
TLDR
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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Journal ArticleDOI
On potentially negative space time covariances obtained as sum of products of marginal ones
TL;DR: This paper proposes an easy-to-implement and interpretable class of models that admits this type of covariances and shows particular analytical examples that may be of interest in the biometrical context.
Journal ArticleDOI
Complete monotonicity of a function involving the ratio of gamma functions and applications
Feng Qi,Chun-Fu Wei,Bai-Ni Guo +2 more
TL;DR: In this article, necessary and sucient conditions are presented for a function involving a ratio of gamma functions to be logarithmically com- pletely monotonic.
Certain Logarithmically N-Alternating Monotonic Functions Involving Gamma and q-Gamma Functions
TL;DR: In this paper, three basic properties of logarithmically N-alternating monotonic functions are established and the monotonicity results of some functions involving the gamma and q-gamma functions, which are obtained by W. E. Clark and M. H. Ismail.
Journal ArticleDOI
Bounds for the ratio of two gamma functions--From Gautschi's and Kershaw's inequalities to completely monotonic functions
TL;DR: In this paper, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma functions was analyzed, and some new bounds for the ratio of two Gamma functions and divided differences of polygamma functions were given.
Journal ArticleDOI
Sharp inequalities for polygamma functions
Feng Qi,Bai-Ni Guo +1 more
TL;DR: In this article, it was shown that the double inequalities of the gamma, psi and polygamma functions are monotonically monotonicity results, and that the constants of the Gamma, psi, and polygonal functions are the best possible.
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