On Some Semi-Classical Properties of Perturbations of a Second-Order Self-Associated Orthogonal Polynomial Sequence

Second-order self-associated orthogonal polynomial sequences are essential cases of second-degree and semi-classical families. In this work, we take as a starting point a case of those sequences, and we perturb it by modifying one of its recurrence coefficients by translation or by dilation at an arbitrary order. In this way, we generate new semi-classical orthogonal families. The goal of this paper is to study such perturbed sequences. For that, we apply a general method for deriving some semi-classical properties of perturbed second-degree forms, and we use the symbolic implementation of that method in the Mathematica ® language to obtain some properties of such families, namely their Stieltjes functions, Stieltjes equations, their classes, structured relations, and second-order differential equations. We give new explicit results for the first orders of perturbations. Also, we highlight some particular cases.