Orthogonal Polynomials Associated with Some Modification of a Linear Form

Abstract

We show that if v is a regular Laguerre-Hahn (resp. semiclassical) linear form,then the linear form u defined by the relation LAMBDA(x - a )u = (x - c)v is also regular and a Laguerre-Hahn (resp. semiclassical) linear form for every complex LAMBDA except for a discrete set of numbers depending on v, a, and c. We give explicitly the coefficients of the three-term recurrence relation, the structure relation of the orthogonal sequence associated with u, and the class of the linear form u knowing that of v. Finally, we apply the above results to some examples.