A note on multidimensional modified fractional calculus operators
type: Item
instance of: scholarly article
title: A note on multidimensional modified fractional calculus operators
zbMATH Open document ID: 0876.26010
DOI: 10.1007/BF02837168
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1997-11-20T00:00:00Z
review text: For functions \(f(x)\), \(x\in\mathbb{R}^n_+\), there exist many definitions of what is fractional integration or differentiation in case of many variables. Apart from obvious definitions of partial or mixed fractional derivatives and integrals, there exist essentially multidimensional versions, such as fractional differentiation in a given direction, fractional differentiation of elliptic, hyperbolic or parabolic type and others [see the book by the reviewer and \textit{A. A. Kilbas} and \textit{O. I. Marichev}, ``Fractional integrals and derivatives. Theory and applications'' ((Russian) 1987; Zbl 0617.26004, (English translation) 1993; Zbl 0818.26003), Sections 24-29]. In \textit{Yu. A. Brychkov}, \textit{H.-J. Glaeske}, \textit{A. P. Prudnikov} and \textit{V. K. Tuan}: ``Multidimensional integral transformations'' (1992; Zbl 0752.44004) there was suggested a definition of the fractional integration in the form \[ X^\alpha_+f(x)= {1\over\Gamma(\alpha+ 1)} \sum^n_{k=1} {\partial\over\partial x_k} x^{1- n}_k \int^{x_k}_0 (x_k- t)^{\alpha}t^{n-\alpha- 1}f\Biggl({t\over x_k} x\Biggr) dt,\quad x\in\mathbb{R}^n_+, \] and a similar form \(X^\alpha_-f\) with the integration over \((x_k,+\infty)\), where it is evidently assumed that \(0<\text{Re }\alpha<n\). The author derives some formulas for calculation of such integrals \(X^\alpha_{\pm}f\) when \(f\) is expressed in a certain way in terms of the \(H\)-function of Fox, the final formulas being too complicated to be reproduced here. The action of the operators of the type \(x^\beta X^\alpha_{\pm}x^{-\beta}\) is also studied in some special spaces \(M_\gamma(\mathbb{R}^n_+)\) defined via analyticity properties of its functions \(f(x)\) after the substitution \(x\to e^x\). Reviewer's remark: By easy calculations \(X^\alpha_+f(x)\) is reduced to \[ X^\alpha_+f(x)= {1\over\Gamma(\alpha)} \int^1_0 t^{n-\alpha-1}(1- t)^{\alpha-1}f(tx)dt \] and similarly for \(X^\alpha_-f(x)\), which should simplify calculations. We remark also…
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 955218
zbMATH Keywords: multidimensional fractional calculus
zbMATH Keywords: Riemann-Liouville fractional integrals
zbMATH Keywords: Fox \(H\)-function
zbMATH Keywords: Mellin transform
author: Ravinder Krishna Raina
reviewed by: S. G. Samko
MaRDI profile type: Publication
cites work: Q4000492
cites work: The G and H Functions as Symmetrical Fourier Kernels
cites work: Q4274274
cites work: Q4161583
cites work: Q3141902
cites work: Q3999270
cites work: On composition of certain fractional integral operators
cites work: Q4020154
cites work: Q3874511
cites work: Q3663556
cites work: Q4280531
cites work: Q4244512
cites work: Q3141895
cites work: Q4743083
cites work: Q3475815
cites work: Multidimensional Modified Fractional Calculus Operators
description: scientific article; zbMATH DE number 955218