Third-degree semiclassical forms of class one arising from cubic decomposition

Abstract

ABSTRACT An orthogonal polynomial sequence with respect to a regular form (linear functional) w is said to be semiclassical if there exists a monic polynomial φ and a polynomial ψ with , such that . Recently, all semiclassical monic orthogonal polynomial sequences , of class one obtained from the cubic decompositions (CD) satisfying the relation have been determined [see Tounsi MI, Bouguerra I. Cubic decomposition of a family of semiclassical polynomial sequences of class one. Integral Transforms Spec Funct. 2015;26(5):377–394; Castillo K, de Jesus MN, Petronilho J. On semiclassical orthogonal polynomials via polynomial mappings. J Math Anal Appl. 2017;455(2):1801–1821]. The aim of our work is to study semiclassical sequences of the above family such that their corresponding Stieltjes function satisfies a cubic relation of the form , where A, B, C, D are polynomials. In particular, the link between w and the Jacobi form is established. Furthermore, both the characteristic elements of the structure relation and of the second-order differential equation are explicitly given.