An Abelian theorem for a general class of Mellin-type integral transforms
type: Item
instance of: scholarly article
title: An Abelian theorem for a general class of Mellin-type integral transforms
zbMATH Open document ID: 0647.46040
DOI: 10.1016/0022-247X(88)90049-2
author: Paul Embrechts
author: Edward Omey
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The reviewer established some Abelian theorems for the distributional H- function transform [Acta Mexicana C. Technol. 7, 66-76 (1973; Zbl 0313.44005)] associated with an H-function of \textit{Charles Fox} [Trans. Am. Math. Soc. 98, 395-429 (1961; Zbl 0096.308)] in the form \[ (1)\quad F(s)=\int^{\infty}_{0}\exp (-\omega ts)H^{m,n}_{p,q}\left[\rho (st)^{\mu}\left| \begin{matrix} (a_ p,\alpha_ o)\\ (b_ q,\beta_ q)\end{matrix} \right]f(t)dt\right. \] for the case that s is real. Here H- function is defined as \[ H^{m,n}_{p,q}\left[z\left| \begin{matrix} (a_ p,\alpha_ p)\\ (b_ q,\beta_ q)\end{matrix} \right]\right. =1/2\pi 1\int_{C}\hat H(s)z^ s ds, \] where \[ (2)\quad \hat H(s)=\frac{\prod^m_{j=1}\Gamma (b_ j-\beta_ js)\prod^{n}_{j=1}\Gamma (1-a_ j+\alpha_ js)}{\prod^{q}_{j=m+1}\Gamma (1-b_ j+\beta_ js)\prod^{p}_{j=n+1}\Gamma (a_ j-\alpha_ js)}. \] A detailed account of the H-function with applications in statistics and other disciplines can be found in a monograph by \textit{A. M. Mathai} and the reviewer [The H-function with applications in statistics and other disciplines (1978; Zbl 0382.33001)]. For \(n=0\), the H-transform was discussed by \textit{V. G. Joshi} and \textit{R. K. Saxena} [Math. Ann. 256, 311-321 (1981; Zbl 0478.46044)], where they also proved certain abelian theorems. Recently, \textit{Richard D. Carmichael} and \textit{Ram S. Pathak} [Math. Proc. Camb. Philos. Soc. 102, 533-552 (1987; Zbl 0632.44002)] also established Abelian theorems for the H-transform (1) of functions and generalized functions as the complex variable of the transform approaches zero or infinity in a wedge domain in the right half plane. In order to unify and extend the results of the reviewer and of V. G. Joshi and R. K. Saxena (Indore) referred above, the authors consider a class of functions which are closely connected with the class of regularly varying functions introduced by Karamata [cf. \textit{E. Seneta}, Regularly varying functions (1976; Zbl 0324.26002)]. The authors…
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 4056321
zbMATH Keywords: Mellin-type integral transforms
zbMATH Keywords: regularly varying functions
zbMATH Keywords: Abelian theorems for the distributional H-function transform
MaRDI profile type: Publication
cites work: Q4186817
cites work: Mercerian and Tauberian theorems for differences
cites work: Q5633348
cites work: An Abel-Tauber Theorem for Laplace Transforms
cites work: Convolution inequalities, regular variation and exceptional sets
cites work: The G and H Functions as Symmetrical Fourier Kernels
cites work: Regularly Varying Functions and Convolutions with Real Kernels
cites work: Abelian theorems for distributional H-transform
cites work: Q4161583
cites work: Some Abelian theorems for distributional Meijer Laplace transformation
cites work: On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin
cites work: Q4072303
cites work: Regularly varying functions in the theory of simple branching processes
cites work: Q5824493
cites work: Q4726487
description: scientific article; zbMATH DE number 4056321