On the Second Order Differential Equation Satisfied by Perturbed Chebyshev Polynomials

In some applications one is led to consider perturbations of orthogonal polynomials translated by a modification on the first coefficients of the second order recurrence relation satisfied by these polynomials. Moreover, the four Chebyshev families are among the most useful orthogonal sequences due to their exceptional features. Thus, it is important to clarify and explicit the properties of perturbed Chebyshev polynomials, in special the second order linear differential equation that assures their semi-classical character. This is the main goal of this work. By means of a symbolic algebraic algorithm based on Stieltjes equations, we are able to explicit new properties for the complete perturbation of order one and an special perturbation of order two for all four Chebyshev families.