Characterization of the symmetric D-Laguerre–Hahn orthogonal polynomial sequences of even class via the quadratic decomposition

Abstract

A regular form w is said to be D-Laguerre–Hahn of class the nonnegative integer , if there exist four coprime polynomials (monic), B, C and D (D depends on ) such that the Stieltjes function S(w) satisfies the Riccati differential equation . The class is then equal to . We proceed to the quadratic decomposition of symmetric D-Laguerre–Hahn orthogonal polynomial sequences of even class and we characterize the two derived sequences and their corresponding components u and v obtained by this approach. We describe the class and the Riccati equation of u and v into detail. Some examples from the class two are highlighted. In addition, some D-Laguerre–Hahn forms of class one are discovered.