Higher Order Coherent Pairs
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Higher Order Coherent Pairs Francisco Marcellán · Natalia Camila Pinzón
Received: 6 October 2011 / Accepted: 15 February 2012 / Published online: 1 March 2012 © Springer Science+Business Media B.V. 2012
Abstract In this paper, we study necessary and sufficient conditions for the relation [r] (x) = Rn−r (x) + bn−1,r Rn−r−1 (x), Pn[r] (x) + an−1,r Pn−1
an−1,r = 0, n ≥ r + 1, 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 U , 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
1 Introduction The notion of coherent pair was introduced by A. Iserles, P.E. Koch, S.P. Nørsett and J.M. Sanz-Serna in 1991 [15]. They state that a pair of positive Borel measures (μ0 , μ1 ) supported on the real line is, with our terminology, a (1, 0)-coherent pair of order 1 if and only if there exist nonzero constants {an,1 }n≥1 such that their corresponding sequences of monic F. Marcellán () · N.C. Pinzón Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Madrid, Spain e-mail: [email protected] N.C. Pinzón e-mail: [email protected]
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F. Marcellán, N.C. Pinzón
orthogonal polynomials (SMOP) {Pn (x)}n≥0 and {Rn (x)}n≥0 satisfy Rn (x) =
(x) Pn+1 P (x) + an,1 n , n+1 n
an,1 = 0, n ≥ 1.
(1.1)
Moreover, this condition of coherence is a sufficient condition for the existence of a relation Qn+1 (x; λ) + cn,1 (λ)Qn (x; λ) = Pn+1 (x) +
n+1 an,1 Pn (x), n
n ≥ 1,
(1.2)
where {cn,1 (λ)}n≥1 are rational functions in λ > 0 and {Qn (x; λ)}n≥0 is the SMOP associated with the Sobolev inner product
p(x), q(x) λ,1 =
∞ −∞
p(x)q(x) dμ0 + λ
∞ −∞
p (x)q (x) dμ1 ,
λ > 0, p, q ∈ P.
(1.3)
Besides, they study the case when the measure μ0 is classical (Laguerre and Jacobi). Furthermore, they introduce the notion of symmetrically coherent pair, when the two measures μ0 and μ1 are symmetric (i.e., invariant under the transformation x → −x) and the subscripts in (1.1) are changed appropriately. In 1995, F. Marcellán, T. Pérez, J.C. Petronilho, and M. Piñar (see [20]) showed that if a pair of positive definite linear functionals (U , V ) is a (1, 0)-coherent pair of order 1, then both are semiclassical, V is of class at most 1 and U is of class at most 6. Moreover, τ2 (x) such that σ2 (x)V = τ2 (x)U , with they proved that there exist polynomials σ2 (x) and τ2 (x)) ≤ 3. deg( σ2 (x)) ≤ 2 and deg( On the other hand, F. Marcellán and J. Petronilho [18] studied (1.1) when U and V , with respective SMOP {Pn (x)}n≥0 and {Rn (x)}n≥0 , are
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