A non-symmetric second-degree semi-classical form of class one

Abstract

An orthogonal polynomial sequence with respect to a regular form (linear functional) u is said to be semi-classical if there exist a monic polynomial Φ and a polynomial Ψ, with deg Ψ≥1, such that (Φ u)′+Ψ u=0. Recently, all semi-classical monic orthogonal polynomial sequences of class one satisfying a three-term recurrence relation with β _n =(−1) ⁿ β₀, n≥0, β₀∈ℂ∖{0} have been determined (see [B. Bouras and A. Alaya, A large family of semi-classical polynomials of class one, Integral Transforms Spec. Funct. 18 (2007), pp. 913–931]). In this paper, the sequences of the above family such that their corresponding Stieltjes function S(u)(z)=−∑ _n≥0⟨ u, x ⁿ ⟩/z ⁿ⁺¹ satisfies a quadratic relation of the form BS ²(u)+CS(u)+D=0, where B, C, D are polynomials, are described.

Keywords:

• orthogonal polynomials
• semi-classical forms
• second degree forms

1991 Mathematics Subject Classifications :

• 33C45
• 42C05

Acknowledgements

The work of the third author (FM) has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01. The work of the second author (BB) has been supported by Quassim University, grant SR-D-010-316.