Riesz potential and Riemann-Liouville fractional integrals and derivatives of Jacobi polynomials
type: Item
instance of: scholarly article
title: Riesz potential and Riemann-Liouville fractional integrals and derivatives of Jacobi polynomials
zbMATH Open document ID: 0883.33003
DOI: 10.1016/S0893-9659(96)00119-X
published in: Applied Mathematics Letters
publication date: 1997-10-26T00:00:00Z
review text: The main result is given by the following theorem: ``If \(\alpha>-1\), \(\beta>-1\), \(0<\nu<1\) and \(\gamma\) is an arbitrary real number, then for \(-1< x<1\) holds \[ \begin{multlined} \int_{-1}^1 \Biggl(\text{sign} (x-t)+ \frac{\text{tan}(\pi\Gamma)} {\text{tan} \frac{\nu\pi}{2}} \Biggr) \frac{Q_m^{\alpha,\beta}(t)} {|x-t|^\nu} dt=\\ =\frac{\sin \pi(\gamma-\nu/2) \Phi_1(x)+\sin \pi(\gamma+\nu/2-\beta) \Phi_2(x)} {\Gamma(\nu)\sin \frac{\nu\pi}{2}\cos \gamma\pi} \quad (m=0,1,2,\dots) \end{multlined} \] where \[ \begin{aligned} \Phi_1(x)&=\frac{\Gamma(m+\alpha+1) \Gamma(m+\nu) \Gamma(\beta-\nu+1) (-1)^m} {2^{-\alpha-\beta+\nu-1} \Gamma(m+\alpha+\beta- \nu+2)m!}\cdot\\&\quad\cdot{}_2F_1\Biggl(\begin{aligned} m+\nu,\nu-m-\alpha &-\beta-1\\ &-\beta+\nu\end{aligned} \Biggl|\frac{1+x}{2} \Biggr),\\ \Phi_2(x)&= \frac{\Gamma(m+\beta+1) \Gamma(\nu-\beta-1) (-1)^{m+1}} {2^{-\alpha} (1+x)^{\nu-\beta-1}}\cdot\\ &\quad\cdot{}_2F_1\Biggl(\begin{aligned} m+\beta+1,&-m-\alpha\\ &\beta-\nu+2 \end{aligned} \Biggl|\frac{1+x}{2}\Biggr). \end{aligned} \] Here \(Q_m^{\alpha,\beta}(t)\) stands for the weighted Jacobi polynomial and \({}_2F_1\) the hypergeometric function. By choosing in this formula the parameters appropriately, some spectral relationships for fractional Riesz potentials and for Riemann-Liouville fractional integrals and derivatives of weighted Jacobi polynomials, are obtained as particular cases.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 1078544
zbMATH Keywords: fractional derivative
zbMATH Keywords: weighted Jacobi polynomial
zbMATH Keywords: Riesz potentials
zbMATH Keywords: Riemann-Liouville fractional integrals
author: Igor Podlubny
reviewed by: Nácere Hayek
MaRDI profile type: Publication
cites work: Q5598945
cites work: Q3332356
cites work: The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus
cites work: Q4000098
cites work: Q4726489
cites work: Q3864730
cites work: Q5827353
cites work: Q5847356
cites work: Q4036638
cites work: Singular value decomposition of fractional integration operators in L2-spaces with weights
Recommended article: Extended Jacobi functions via Riemann-Liouville fractional derivative
Recommended article: Q3113610
Recommended article: On the Riemann-Liouville fractional calculus, \(g\)-Jacobi functions and \(F\)-Gauss functions
Recommended article: Q3636013
Recommended article: Liouville and Riemann-Liouville fractional derivatives via contour integrals
Recommended article: Q2935949
Recommended article: Q4528590
Recommended article: Q2987162
Recommended article: Q4700406
Recommended article: Riemann Liouville integrals of fractional order and extended KP hierarchy
description: scientific article; zbMATH DE number 1078544