On the generalized ultraspherical or Gegenbauer functions of fractional orders

摘要

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 Cnλ(x) for n=1,2,…, λ>−(1/2) to functions Cn,αλ(x) where α is any positive number. We prove that this definition generalizes and interpolates the properties of Cnλ(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.