Fractional calculus, Gegenbauer transformations, and integral equations

Starting from the Riemann-Liouville and the Weyl calculus, compositions of fractional integral and fractional differential operators are studied in this paper. These composite operators and their inverses admit descriptions as integral transformations with Gegenbauer functions in their kernel. Rodrigues-type formulas for Gegenbauer functions and new relations for fractional differential and integral operators are derived. Thus classical results on integral equations of Mellin convolution type are extended and unified.