Abstract
In this paper, we study necessary and sufficient conditions for the relation
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
If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, then deg(φ n+r,r (x))≤n+r−1.
If \((\mathcal {U},\mathcal {V})\) is a (1,0)-coherent pair of order r, this equality holds for n≥r+1.
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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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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DOI: https://doi.org/10.1007/s10440-012-9696-0
Keywords
- Coherent pairs
- Sobolev inner product
- Stieltjes functions
- Semiclassical linear functionals
- Orthogonal polynomials