Some symmetric semi-classical polynomial sets (Q2853234)

It is shown that if \(v\) is a regular semi-classical form (linear functional), then the symmetric form \(u\) defined by the relation \(x\sigma u=-\lambda v\), where \(\sigma u\) is the even part of \(u\), is also a regular and semi-classical form for every complex \(\lambda\), except for a discrete set of numbers depending on \(v\). The recurrence coefficients, integral representation and the structure relation coefficients of the orthogonal polynomials sequence associated with \(u\) and the class of the form \(u\) knowing that of \(v\), are given explicitly.