Geodesic
Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function
Fractional calculus and applied analysis, Tome 8 (2005) no. 4, pp. 431-438.

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This paper refers to a fractional order generalization of the classical Jacobi polynomials. Rodrigues’ type representation formula of fractional order is considered. By means of the Riemann–Liouville operator of fractional calculus fractional Jacobi functions are defined, some of their properties are given and compared with the corresponding properties of the classical Jacobi polynomials. These functions appear as a special case of a fractional Gauss function, defined as a solution of the fractional generalization of the Gauss hypergeometric equation.
Keywords: Riemann–Liouville Fractional Differentiation and Integration Operators, Jacobi Polynomials, Rodrigues' Representation, Fractional Jacobi Functions, Gauss Hypergeometric Differential Equation, Fractional Gauss Functions, 26A33, 33C45 
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Gogovcheva, Elena; Boyadjiev, Lyubomir. Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function. Fractional calculus and applied analysis, Tome 8 (2005) no. 4, pp. 431-438. http://geodesic.mathdoc.fr/item/FCAA_2005_8_4_a5/