On an inverse problem for a linear combination of orthogonal polynomials

Abstract

This paper deals with the analysis of the orthogonality of a monic polynomial sequence fQ n } n$0 defined as a linear combination of a sequence of monic orthogonal polynomials fP n } n$0 withwhere r n -0 for n $ 3. Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials fQ n } n$0 . We obtain the connection between the Jacobi matrices associated with fP n } n$0 and fQ n } n$0 , respectively, by using an LU factorization. Some special cases of the above type relation are analysed.

Orthogonality of the Dickson polynomials of the $(k+1)$-th kind

“…Linear combinations of orthogonal polynomials have been studied by many authors; see [3], [35], [42], [43], and [44]. Note that the associated polynomials for both sequences are the same, i.e., q * n (x) = P * n (x).…”

Orthogonal polynomials associated with an inverse spectral transform. The cubic case

“…Let us recall the following result: Theorem 2.1. [8] {Q n } n≥0 is an SMOP if and only if as well asγ 1γ2γ3 0 with s n−1γn = s n γ n−1 + t n (β n−2 −β n ) + r n − r n+1 , n ≥ 2, (2.2) t n−1γn = t n γ n−2 + r n (β n−3 −β n ), n ≥ 3,…”

Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure

“…In [2] the authors provide sharp limits (and the speed of convergence to them) of the zeros of the Geronimus perturbed SMOP, and also, when µ is semi-classical they obtain the corresponding electrostatic model for the zeros of the Geronimus perturbed SMOP, showing that they are the electrostatic equilibrium points of positive unit charges interacting according to a logarithmic potential under the action of an external field. In [17] the authors extend the standard Geronimus transformation to a cubic case. [10] provides a new revision of the Geronimus transformation in terms of symmetric bilinear forms in order to include certain Sobolev and Sobolev-type orthogonal polynomials into the scheme of Darboux transformations.…”