Fractional calculus, i.e. the theory of derivatives and integrals of non-integer orders, enjoys growing interest not only among mathematicians, but also among physicists and engineers (see [J.L. Wenger, F.R. Norwood (Eds.), IUTAM Symposium––Nonlinear Waves in Solids, ASME/AMR, Fairfield, NJ, 1995; J. Alloys Compd. 211/212 (1994) 534; An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons Inc., New York, 1993; On Two Definitions of Fractional Calculus, Slovak Academy of Sciences, Institute of Experimental Physics, UEF-96 ISBN 80-7099-252-2, 1996; B. Ross (Ed.), Fractional Calculus and its Applications, Lecture Notes in Mathematics, vol. 457, Springer-Verlag, Berlin, 1975, p. 1; Phys. Scripta 43 (1991) 174; IEEE Trans. Dielect. Electr. Insulation 1(5) (1994) 826]). In this paper we generalize the Gegenbauer or ultraspherical polynomials C n λ ( x) for n=1,2,…, λ>−(1/2) to functions C n, α λ ( x) where α is any positive number. We prove that this definition generalizes and interpolates the properties of C n λ ( x). The generalized Legendre and Chebyshev polynomials of fractional orders will be studied as special cases. The hypergeometric and R-functions representation will be given.