Riesz potential and Riemann-Liouville fractional integrals and derivatives of Jacobi polynomials
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.
scientific article; zbMATH DE number 1078544
scientific article