Abstract
Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.
Highlights
Introduction and PreliminariesThe theory of orthogonal polynomials constitutes a basic topic in the framework of special functions and approximation theory
Another approach is based on the algebraic analysis of linear functionals and the corresponding sequences of orthogonal polynomials constituting a structural approach where the Stieltjes function associated with the linear functional play a central role
When they satisfy first order linear differential equations with polynomial coefficients, a new hierarchy is stated and you obtain a classification following a different characterization according to the degree of the polynomial coefficients involved in the above differential equation
Summary
The theory of orthogonal polynomials constitutes a basic topic in the framework of special functions and approximation theory. The previous contributions (References [14,15,16]) focus the attention on individual Darboux transformations and their relation with factorizations of the corresponding Jacobi matrices, as well as the connection between the sequences of orthogonal polynomials with respect to such linear functionals. Christoffel perturbations, that appear when considering orthogonality with respect e = p( x )u, where p( x ) is a polynomial, were studied in 1858 to a new linear functional u by E. Notice that the zeros of orthogonal polynomials with respect to a canonical Christoffel transformation (a perturbation by a linear polynomial p( x ) = ( x − c)) of a nontrivial probability measure) are the nodes in the Gauss-Radau quadrature formula. Two illustrative examples about the above questions are discussed
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