Numerical Algorithms

, Volume 60, Issue 2, pp 223–239 | Cite as

(1, 1)-q-coherent pairs

  • Francisco Marcellán
  • Natalia C. Pinzón-Cortés
Original Paper

Abstract

In this paper, we introduce the concept of (1, 1)-q-coherent pair of linear functionals \((\mathcal{U},\mathcal{V})\) as the q-analogue to the generalized coherent pair studied by Delgado and Marcellán in (Methods Appl Anal 11(2):273–266, 2004). This means that their corresponding sequences of monic orthogonal polynomials {Pn(x)}n ≥ 0 and {Rn(x)}n ≥ 0 satisfy
$$ \frac{\left(D_qP_{n+1}\right)(x)}{[n+1]_q} + a_{n}\frac{\left(D_qP_{n}\right)(x)}{[n]_q} = R_{n}(x) + b_{\!n}R_{n-1}(x) \,, \quad\, a_{n}\neq0,\,\, n\geq1, $$
\([n]_q=\frac{q^n-1}{q-1}\), 0 < q < 1. We prove that if a pair of regular linear functionals \((\mathcal{U},\mathcal{V})\) is a (1, 1)-q-coherent pair, then at least one of them must be q-semiclassical of class at most 1, and these functionals are related by an expression \(\sigma(x)\mathcal{U}=\rho(x)\mathcal{V}\) where σ(x) and ρ(x) are polynomials of degrees ≤ 3 and 1, respectively. Finally, the q-classical case is studied.

Keywords

Linear functionals q-orthogonal polynomials q-coherent pairs 

Mathematics Subject Classifications (2010)

42C05 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Francisco Marcellán
    • 1
  • Natalia C. Pinzón-Cortés
    • 1
  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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