A Fractional Integral Operator Corresponding to Negative Powers of a Second Order Partial Differential Operator

摘要

A fractional integral operator is derived for which the partial differential operator plays a role similar to that of ordinary differentiation (d/dx) with respect to the fractional integral of Riemann-Liouville. An application is given to the Koornwinder polynomials (a class of orthogonal polynomials in two variables). Also, an operator is obtained which has the Jacobi functions of order 1/2 (NU-1), -1/2 as its eigenfunctions.