On linearly related orthogonal polynomials and their functionals

Abstract

Let {P n } be a sequence of polynomials orthogonal with respect a linear functional u and {Q n } a sequence of polynomials defined byWe find necessary and sufficient conditions in order to {Q n } be a sequence of polynomials orthogonal with respect to a linear functional v. Furthermore we prove that the relation between these linear functionals is (x −ã)u = λ(x − a)v. Even more, if u and v are linked in this way we get that {P n } and {Q n } satisfy a formula as above.

Companion Linear Functionals and Sobolev Inner Products: A Case Study

“…Condition (1.8) for the pair (L 2,3 ) follows in a similar way that in case (J 2,3 ), because the linear functional V satisfies the conditions studied in [1].…”

“…For example, in (J 1,2 ) we have (x − ξ 1 )U = (x − ξ 0 )J (α+1,β+1) . Then, the results shown in [1] give us a 2 − 2 relation between the polynomials P n and the Jacobi polynomials P (α+1,β+1) n . Taking into account that R n = P (α,β) n as well as the differentiation formula for the Jacobi polynomials, R ′ n = nP (α+1,β+1) n−1 , then it is easy to prove that the pair (J 1,2 ) satisfies a relation of the type (1.8).…”

“…In the above cases, the linear functional V satisfies the conditions described in [1]. For example, in (J 1,2 ) we have (x − ξ 1 )U = (x − ξ 0 )J (α+1,β+1) .…”

“…This problem has been solved in [1] in the context of inverse problems for orthogonal polynomials. In [3] and [4] some particular examples of (1.6) when V is a polynomial perturbation of degree 1 of the Laguerre or Jacobi linear functional are studied.…”

“…In case (J 2,3 ), the linear functional V satisfies the conditions described in [1], that is, (x − ξ 1 )V = (x − ξ 0 )J (α,β) . Then, as the authors show in [1], we have a 2 − 2 relation for R n and the Jacobi polynomials P (α,β) n , from where we obtain directly that this pair verifies (1.8).…”