On the Riemann-Liouville fractional calculus, g-Jacobi functions and F-Gauss functions

This paper refers to a fractional extension of the classical Jacobi polynomials. A fractional order Rodrigues' type representation formula is considered. By means of the Riemann-Liouville operator of fractional calculus, new g-Jacobi functions are defined, some of their properties are given and compared with the corresponding properties of the classical Jacobi polynomials. Furthermore, the hypergeometric equation of Gauss is extended to a fractional order. A new F-Gauss hypergeometric function is defined as a solution to the extended fractional differential equation and considered as a candidate for a fractional hypergeometric function of Gauss.