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Mellin-Barnes integrals for stable distributions and their convolutions. (English) Zbl 1175.26017

The paper is of expository nature. The authors give a survey of Mellin-Barnes integrals, including historical comments. Such integrals arising under application of Mellin transforms, have integrands in the form of a product of quotients of gamma functions, and are known to be used in the definition and/or characterization of higher transcendental functions such as higher hypergeometric functions \(_pF_q\), Meyer \(G\)-function and Fox \(H\)-function.
The authors, in particular, expose the representation of an \(\alpha\)-stable process with Levy index \(\alpha\in (0,2]\) in the probability theory in terms of Mellin transforms and Mellin-Barnes integrals. Special attention is paid to probability densities obtained through a convolution of two stable densities with different indices \(\alpha_1\) and \(\alpha_2\). Finally the authors treat the case \(\alpha_1=1\) and \(\alpha_2=2\) which corresponds to the case of the so called Voight profile.

MSC:

26A33 Fractional derivatives and integrals
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
44A35 Convolution as an integral transform
60G18 Self-similar stochastic processes
60G52 Stable stochastic processes
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