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

作者：Alaya Atef; Bouras Belgacem; Marcellan Francisco^*

来源：Integral Transforms and Special Functions, 2012, 23(2): 149-159.

DOI：10.1080/10652469.2011.567388

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

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 Phi and a polynomial Psi, with deg Psi %26gt;= 1, such that (Phi u)%26apos; + Psi u = 0. Recently, all semi-classical monic orthogonal polynomial sequences of class one satisfying a three-term recurrence relation with beta(n) = (-1)(n)beta(0), n %26gt;= 0, beta(0) is an element of 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) = -Sigma(n%26gt;0)%26lt; u, x(n)%26gt;/z(n+1) satisfies a quadratic relation of the form BS2(u) + CS(u) + D = 0, where B, C, D are polynomials, are described.

• 出版日期2012