Orthogonal polynomials on the semicircle were introduced by Gautschi and Milovanović in [Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (July 1985), pp. 179-185] [J. Approx. Theory, 46 (1986), 230-250]. In this paper we give an account of kind orthogonality, weighted generalizations mainly oriented to Chebyshev weights first second kinds, including several interesting properties such pol...

In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix of fractional derivative of order $ gamma $ in the Caputo for FCSs and show that this matrix with the Tau method are utilized to reduce the solution of some fractional-order differential equations.

Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalan-type Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2) (2x) and (1/2) (4x) in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representa...