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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Some Function Classes Connected to the Class of Completely Monotonic Functions
TL;DR: In this paper, the authors introduce some function classes connected to the class of completely monotonic functions, which they call monotonically monotone functions (Monotonic function classes).
A class of logarithmically completely monotonic functions related to (1 + 1/x) x and an application 1
Da-Wei Niu,Jian Cao,Feng Qi +2 more
TL;DR: In this paper, sufficient and necessary conditions such that the function [ e/(1 + 1/x)x+β ]x+α is completely monotonic or logarithmically complete in (0,∞) are established.
Posted Content
Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function
Feng Qi,Mark Daniel Ward +1 more
TL;DR: In this article, the authors established Taylor's series expansions of several functions involving the inverse tangent function, and analyzed some properties, including generating functions, limits, positives, negatives, monotonicity, and logarithmic convexity, of the coefficients in the series expansion of Wilf's function.
Journal ArticleDOI
Logarithmically completely monotonic functions related to the q-gamma function and its applications
Khaled Mehrez,Sourav Das +1 more
Journal ArticleDOI
Some Problems Related to Completely Monotone Positive Definite Functions
TL;DR: In this article, it was shown that if f ∈ C ∞(0, + ∞) and δ ∈ ℝ, then if f\left( x \right) - {a^\delta }f( {ax} \right)\; \in \;{\mathcal C}{\mathcal M}\;\;\,\;
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