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
β
0
, n≥0, β
0
∈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.