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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Some Bernstein functions and integral representations concerning harmonic and geometric means
TL;DR: In this article, the authors show that the harmonic mean and the geometric mean are all Bernstein functions of √ √ n, where n is the number of nodes in the graph.
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A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder
TL;DR: A new asymptotic expansion of a ratio of two gamma functions is established as aninline-formula content-type="math/mathml" as well as anotation encoding="application/x-tex" for right-arrow normal infinity.
Journal ArticleDOI
Integral Representations of the Large and Little Schröder Numbers
TL;DR: In this paper, the authors established several integral representations for the generating functions of the large and little Schroder numbers and for the small and the large Schroder number for generating functions.
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Complete monotonicity of divided differences of the di- and tri-gamma functions with applications
Feng Qi,Bai-Ni Guo +1 more
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Monotonicity properties for a ratio of finite many gamma functions.
Feng Qi,Feng Qi,Dongkyu Lim +2 more
TL;DR: In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as completemonotonicity, the Bernstein function property, and logarithmically complete monotonicism.
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