A characterization of the four Chebyshev orthogonal families

We obtain a property which characterizes the Chebyshev orthogonal polynomials of first, second, third, and fourth kind. Indeed, we prove that the four Chebyshev sequences are the unique classical orthogonal polynomial families such that their linear combinations, with fixed length and constant coefficients, can be orthogonal polynomial sequences. 1. Introduction The classical orthogonal polynomials (OP) on the real line, which are the most useful and important families, can be characterized by different conditions (see [3]). In the case of bounded support, the classical model is the family of Jacobi polynomials, which has been extensively studied. Very well-known families of Jacobi polynomials are the so-called Chebyshev polynomials of first, second, third, and fourth kind (see [9]). The orthogonal polynomials with respect to rational modifications of these Chebyshev weights are the Bernstein polynomials (see [3, 9]). It is well known that the Bernstein polynomials can be expressed as linear combinations of Chebyshev polynomials, with fixed length and constant coefficients (see [4, 5]). This property of orthogonality satisfied by the linear finite combinations of sequences of orthogonal polynomials is quite general. Indeed, taking into account the result of Uvarov (see [10]), the orthogonal polynomial sequences with respect to a rational modification of a positive measure on an interval can be written as a linear combination of the orthogonal polynomials with respect to the original measure, with fixed length and coefficients depending on the degrees. Although the representation of the Bernstein polynomials is a consequence of Uvarov's result, it is important to note that in this first case, the coefficients are constant. The study of linear combinations of orthogonal polynomials is an interesting subject for different reasons. For example, several quadrature formulas have been obtained choosing the nodal points as the zeros of some linear combinations. Another reason could be the important role that these linear combinations of OP play in the problem of lin-earization of products of OP, as well as, in the problem of approximation of the polynomial solution for differential equations. For a detailed presentation of these problems and some others, see [6].