Acta Applicandae Mathematicae

, Volume 121, Issue 1, pp 105–135

Higher Order Coherent Pairs

Article

DOI: 10.1007/s10440-012-9696-0

Cite this article as:
Marcellán, F. & Pinzón, N.C. Acta Appl Math (2012) 121: 105. doi:10.1007/s10440-012-9696-0

Abstract

In this paper, we study necessary and sufficient conditions for the relation
$$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$
where {Pn(x)}n≥0 and {Rn(x)}n≥0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals \(\mathcal{U},\mathcal{V}\), respectively, or associated with two positive Borel measures μ0,μ1 supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.

Keywords

Coherent pairs Sobolev inner product Stieltjes functions Semiclassical linear functionals Orthogonal polynomials 

Mathematics Subject Classification (2000)

42C05 

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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