On the generalized ultraspherical or Gegenbauer functions of fractional orders (Q1826668)

The author introduce generalized ultraspherical functions of fractional orders \(C^\lambda_{n,\alpha} (x)\) where \(\alpha\) is a positive real number, \(1>- (\tfrac 12)\) and prove that these generalize and interpolate the known ultraspherical or Gegenbauer polynomials \(C^\lambda_n(x)\), \(n=1,2,\dots\); \(\lambda>- (\tfrac 12)\). Some properties of the generalized Legendre and Chebyshev polynomials of fractional orders have been studied. Also the hypergeometric and \(R\)-functions representation of this function is given.