This well organized paper starts with discussions of the Weyl and the Riemann-Liouville fractional calculi. A number of identities which connect these fractional integral operators with the Erdélyi-Kober and other operators are developed. The Rodrigues formula for Gegenbauer polynomials is then generalized to an integral relation for Gegenbauer functions.
Integral transformations with kernels which involve Gegenbauer functions are then introduced. Numerous operator identities are developed which connect the Gegenbauer integral operators with the previously introduced operators. These identities allow the factorization of the Gegenbauer integral operators into simpler fractional integral and Erdélyi-Kober operators so that the existence and uniqueness of the solution of related integral equations can be obtained. [Note: Equations (26) and (29) show that it is ultimately no restriction to eliminate the parameter by setting \(\mu =-1\).]Solutions to a number of special cases of the integral equations from the literature are then efficiently obtained.
For extensive related work and additional references on this subject, see S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and derivatives of fractional order and some of their applications (Minsk, 1987; Zbl 0617.26004, Engl. transl. Zbl 0818.26003), especially Section 35 and the connected materials.