Quadratic Decomposition of a Laguerre-Hahn Polynomial Sequence II

In 4] the authors proved that a quasi-symmetric orthogonal polynomial sequence {R_n}_{n 3 0}{{\{R_{n}\}}_{n \ge 0}} is a Laguerre-Hahn sequence if and only if the component {P_n}_{n 3 0}{{\{P_{n}\}}_{n \ge 0}} in its quadratic decomposition is also a Laguerre-Hahn sequence. In this paper and under these conditions, we deduce the class s of the Laguerre-Hahn sequence {R_n}_{n 3 0}{{\{R_{n}\}}_{n \ge 0}}. More precisely, if s′ is the class of {P_n}_{n 3 0}{{\{P_{n}\}}_{n \ge 0}} then 2s′ ≤ s ≤ 2s′ + 3. On the other hand the polynomial coefficients of the Riccati equation satisfied by the Stieltjes function corresponding to {R_n}_{n 3 0}{{\{R_{n}\}}_{n \ge 0}} are given in terms of those of {P_n}_{n 3 0}{{\{P_{n}\}}_{n \ge 0}}. As an application, we determine all non-symmetric quasi-symmetric Laguerre-Hahn sequences of class one.