In this section, we give new results, obtained with

\left. . From them, we obtain directly the closed formulas of the coefficients of the differential equation , our main goal in this work. In the particular case of Chebyshev sequences, we are able to compute closed formulas of the generating functions for any perturbation using (2.8).

In this section, we give new results, obtained with , for the complete perturbation of order 1 and the following perturbation of order two of the second kind Chebyshev form
or , from which we can easily obtain the results for the other three Chebyshev families as follows;
,