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.

中文翻译：

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三次分解产生的第一类的三次半经典形式

摘要 如果存在一个一元多项式 φ 和一个多项式 ψ ，使得 ，那么关于正则形式（线性泛函）w 的正交多项式序列被称为半经典的。最近，从满足该关系的三次分解 (CD) 中获得的第一类的所有半经典单调正交多项式序列 已确定 [参见 Tounsi MI, Bouguerra I. 第一类半经典多项式序列族的三次分解。积分变换规范功能。2015;26(5):377–394; Castillo K, de Jesus MN, Petronilho J. 关于通过多项式映射的半经典正交多项式。J数学肛门应用。2017;455(2):1801–1821]。我们工作的目的是研究上述家族的半经典序列，使得它们对应的 Stieltjes 函数满足形式为 的三次关系，其中 A、B、C、D 是多项式。特别地，建立了 w 和雅可比形式之间的联系。此外，明确给出了结构关系和二阶微分方程的特征元素。