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

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 $\mathrm{deg}\psi \ge 1$, such that $(\phi w{)}^{\prime }+\psi w=0$. Recently, all semiclassical monic orthogonal polynomial sequences $\{W_n{\}}_{n\ge 0}$, of class one obtained from the cubic decompositions (CD) satisfying the relation $W_{3n}\left(x\right)=P_n(x^3+qx+r)$ 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 $S\left(w\right)\left(z\right)=-\sum _{n\ge 0}⟨w,x^n⟩/z^{n+1}$ satisfies a cubic relation of the form $A{S}^3\left(w\right)+B{S}^2\left(w\right)+CS\left(w\right)+D=0$, where A, B, C, D are polynomials. In particular, the link between w and the Jacobi form $\mathcal{V}=\mathcal{J}(-2/3,-1/3)$ is established. Furthermore, both the characteristic elements of the structure relation and of the second-order differential equation are explicitly given.

Keywords:

• Orthogonal polynomials
• semiclassical forms
• Stieltjes function
• cubic decomposition
• third-degree forms
• differential equations

AMS classifications:

• Primary: 33C45
• Secondary: 42C05

Disclosure statement

No potential conflict of interest was reported by the author(s).