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Higher Order Coherent Pairs

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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 {P n (x)} n≥0 and {R n (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 μ 0,μ 1 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.

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Notes

  1. If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, then deg(φ n+r,r (x))≤n+r−1.

  2. In (1,0)-coherence of order r, (3.7) and (3.15) coincide.

  3. If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, this equality holds for n≥r+1.

  4. If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, then the polynomial φ n+r,r (x) has degree at most n+r−1 and its expression corresponds to φ n+r−1,r (x) in (4.3).

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Acknowledgements

The authors thank the referees for the careful revision of the manuscript. Their comments and suggestions contributed to improve its presentation. The work of the first author (FM) has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain under grant MTM2009-12740-C03-01.

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  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain

    Francisco Marcellán & Natalia Camila Pinzón

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  1. Francisco Marcellán
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  2. Natalia Camila Pinzón
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Correspondence to Francisco Marcellán.

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Marcellán, F., Pinzón, N.C. Higher Order Coherent Pairs. Acta Appl Math 121, 105–135 (2012). https://doi.org/10.1007/s10440-012-9696-0

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