Fractional Chebyshev differential equation on symmetric \alpha dependent interval‎

Most of fractional differential equations are considered on a fixed interval. In this paper, we consider a typical fractional differential equation on a symmetric interval [-\alpha,\alpha], where \alpha is the order of fractional derivative. For a positive real number α we prove that the solutions are  T_{n,\alpha}(x)=(\alpha+x)^\frac{۱}{۲}Q_{n,\alpha}(x) where Q_{n,\alpha}(x) produce a family of orthogonal polynomials with respect to the weight functionw_\alpha(x)=(\frac{\alpha+x}{\alpha-x})^{\frac{۱}{۲}} on [-\alpha,\alpha]. For integer case \alpha = ۱ , we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional differential equations.

Keywords:

Orthogonal polynomials , Fractional Chebyshev differential equation , Riemann-Liouville and Caputo derivatives