Fractional equivalent of fourth-order Laguerre and Chebyshev polynomials

Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second-order DE. In this paper, we develop a fractional analog of the fourth-order Laguerre differential equation on the interval [0, ∞) and, in addition, we develop a fractional analog of the fourth-order Chebyshev differential equation on a symmetric interval [−α, α], where α is the order of fractional derivative. Using Riemann-Liouville and Caputo fractional derivatives we derive the general theoretical facts concerning the exact form of the polynomials, their orthogonality, and the recursion relations for their coefficients. In the classical case, Laguerre equations of the second and fourth-order have similar solutions with different eigenvalues, Chebyshev equations of the second and fourth order also have similar solutions with different eigenvalues, we discovered similar relations for the fractional forms of these equations.