On modified kernel polynomials and classical type Sobolev orthogonal polynomials

Abstract

In this paper we study modified kernel polynomials: $u_n(x)={\sum }_{k=0}^n{c}_k{g}_k(x)$, depending on parameters $c_k>0$, where $\{g_k{\}}_0^{\infty }$ are orthonormal polynomials on the real line. Besides kernel polynomials with $c_k=g_k(t_0)>0$, for example, $c_k$ may be chosen to be some other solutions of the corresponding second-order difference equation of $g_k$. It is shown that all these polynomials satisfy a $4$-th order recurrence relation. The cases with $g_k$ being Jacobi or Laguerre polynomials are of a special interest. Suitable choices of parameters $c_k$ imply $u_n$ to be Sobolev orthogonal polynomials with a $(3\times 3)$ matrix measure. Moreover, a further selection of parameters gives differential equations for $u_n$. In the latter case, polynomials $u_n(x)$ are solutions to a generalized eigenvalue problems both in $x$ and in $n$.

15 pages

References in corpus (2)

• The semiclassical--Sobolev orthogonal polynomials: a general approach
• The Inverse Spectral Problem for Jacobi-Type Pencils

Cited by in corpus (1)

• On the generalized hypergeometric function, Sobolev orthogonal polynomials and biorthogonal rational functions