Is a pair of polynomials, such that

where is a pair of polynomials, such that is monic, , and . The class of is the minimum integer that satisfies . The class of a form is unique, but and are not 27 . The sequence orthogonal with respect to is also called semi-classical of class s. In two operations on appear 26 : the left product of a form by a polynomial and the derivative of a form defined by transposition from the corresponding operations on as and .
Semi-classical forms are particular cases of Laguerre-Hahn forms . A form is a Laguerre-Hahn if it is regular and its formal Stieltjes function satisfies the so-called Stieltjes equation

where and are polynomials. The sequence orthogonal with respect to is also called a Laguerre-Hahn sequence. If , then the form is called a second degree form (SD) 27,30 . If and supposed monic, we let . Then if , the form is called a strict Laguerre-Hahn form, whereas if , the form is semi-classical. In this last case, is equivalent to the following equation [27]