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

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.

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