Generating Some Symmetric Semi-classical Orthogonal Polynomials

We show that if v is a regular semi-classical form (linear functional), then the symmetric form u defined by the relation x2σu = -λv, where (σƒ )(x) = f (x2) and the odd moments of u are 0, is also regular and semi-classical form for every complex λ except for a discrete set of numbers depending on v. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with u and the class of the form u knowing that of v. We conclude with an illustrative example.

Some symmetric semi-classical polynomial sets

We show that if ? is a regular semi-classical form (linear functional), then the symmetric form u defined by the relation x?u=??? where ?u is the even part of u, is also regular and semi-classical form for every complex ? except for a discrete set of numbers depending on ?. We give explicitly 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 ?. We conclude with some illustrative examples.