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 [-α;α], where α is the order of fractional derivative. For a positive real numberα we prove that the solutions are Tn;α(x) = (α + x)1 2 Qn;α(x), where Qn;α(x) produce a family of orthogonal polynomials with respect to the weight function wα(x) = (α+x -αx ) 1 2 on [-α;α]. For integer c seα = 1, 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.