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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Bounds for the ratio of two gamma functions.
TL;DR: A long history of bounding the ratio for and, various origins of this topic are clarified, several developed courses are followed, different results are compared, useful methods are summarized, new advances are presented, some related problems are pointed out, and related references are collected as discussed by the authors.
Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem
TL;DR: A long history of bounding the ratio for and, various origins of this topic are clarified, several developed courses are followed, different results are compared, useful methods are summarized, new advances are presented, some related problems are pointed out, and related references are collected as mentioned in this paper.
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Completely monotonic functions involving the gamma and q-gamma functions
TL;DR: In this paper, an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic were given. And they were shown to give an infinitely divisible probability distribution for specific combinations of the gamma and q-gamma functions.
Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity
TL;DR: A survey of the relation between positive definite and negative definite functions on abelian semigroups with involution can be found in this paper, where it is shown that the Gamma function is logarithmically completely monotonic.
References
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Journal ArticleDOI
Some classes of completely monotonic functions, II
Horst Alzer,Christian Berg +1 more
TL;DR: In this article, the gamma, digamma, and polygamma functions are defined in terms of the classical gamma functions, and a new class of completely monotonic functions is presented.
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On infinitely divisible matrices, kernels, and functions
TL;DR: In this article, the authors discuss two applications to the study of characteristic functions and completely monotonic functions, and show how the classical representation theorems for infinitely divisible laws may be obtained.
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A completely monotone function related to the Gamma function
TL;DR: In this paper, it was shown that the reciprocal of the function f(z)= log Γ(z+1) z log z, z∈ C ⧹]−∞,0] is a Stieltjes transform.
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Pick Functions Related to the Gamma Function
TL;DR: In this paper, it was shown that the function f(z) = log Γ(z+1)/zlog 2, holomorphic in the complex plane cut along the negative real axis, is a Pick function and its integral representation was given.