An Abelian theorem for a general class of Mellin-type integral transforms
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 establish two theorems which depict the asymptotic behaviour of the H- transform of functions that behave like a regularly varying function at infinity or at zero.
Abelian theorems for the distributional H-function transform
scientific article; zbMATH DE number 4056321