# Bases of the space of solutions of some fourth-order linear difference equations: applications in rational approximation Articles

## Overview

### published in

### publication date

- October 2013

### start page

- 1632

### end page

- 1644

### volume

- 19

### Digital Object Identifier (DOI)

### full text

### International Standard Serial Number (ISSN)

- 1023-6198

### Electronic International Standard Serial Number (EISSN)

- 1563-5120

### abstract

- It is very well known that a sequence of polynomials {Q(n)(x)}(n=0)(infinity) orthogonal with respect to a Sobolev discrete inner product < f; g >(s) = integral(I)fg d mu + lambda f(-1)(0)g'(0); lambda is an element of R+; where mu is a finite Borel measure and I is an interval of the real line, satisfies a five- term recurrence relation. In this contribution we study other three families of polynomials which are linearly independent solutions of such a five- term linear difference equation and, as a consequence, we obtain a polynomial basis of such a linear space. They constitute the analogue of the associated polynomials in the standard case. Their explicit expression in terms of {Q(n)(x)}(n=0)(infinity) using an integral representation is given. Finally, an application of these polynomials in rational approximation is shown.