On generating functions of some perturbations of a second-order self-associated orthogonal sequence

In this work, we take as a starting point a second-order self-associated orthogonal polynomial sequence with periodic recurrence coefficients, which is of second-degree and semi-classical of class one. We consider perturbations of this remarkable sequence obtained by modifying one of its recurrence coefficients, either by translation or by dilation, at a fixed order r. In this way, we generate new semi-classical families. Based on the corresponding recurrence relations, we derive a general closed formula for the generating functions of these perturbed sequences. The symbolic implementation of this formula makes it possible to compute the generating functions for any fixed r. Explicit formulas are obtained for the first few orders of perturbation.