Perturbed recurrence relations

In this paper, we give some new relations between two families of polynomials defined by three-term recurrence relations. These relations allow us to study how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation are perturbed. In the literature some methods are already available. However, most of them are only effective for small perturbations. In order to show the sharpness of our method, we compare it with Gronwall's classical method in the case of large perturbations. Using our tool, we also give a relation between the differential equation satisfied by a family of orthogonal polynomials and its perturbed family. Some explicit results are obtained for Chebyshev polynomials of the second kind.