Second structure relation for semiclassical orthogonal polynomials

Marcellán, Francisco; Sfaxi, Ridha

doi:10.1016/j.cam.2006.01.007

Cited by 13 publications

(9 citation statements)

References 13 publications

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“…Now, we recall and generalize some results concerning the finite‐type relations. It can be considered and used to establish characterizations of classical and semi‐classical polynomials , , , . First, let

{\{P_{n}\}}_{n \geq 0}

and

{\{R_{n}\}}_{n \geq 0}

be sequences of monic polynomials with

{\{u_{n}\}}_{n \geq 0}

and

{\{w_{n}\}}_{n \geq 0}

their respective dual sequences, the following formula always holds:

ϕ (x) R_{n} (x) = \sum_{ν = 0}^{n + t} λ_{n, v} P_{ν} (x), n \geq 0,

where

λ_{n, v} = 〈u_{v}, ϕ R_{n}〉,

0 \leq v \leq n + t,

n \geq 0 .

Definition When there is an integer

s \geq 0

such that we shall say that is a finite‐type relation between

{\{P_{n}\}}_{n \geq 0}

and

{\{R_{n}\}}_{n \geq 0}

, with respect to ϕ. Lemma For any sequence

{\{P_{n}\}}_{n \geq 0}

compatible with ϕ,

deg ϕ <...>

…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Now, we recall and generalize some results concerning the finite-type relations. It can be considered and used to establish characterizations of classical and semi-classical polynomials [34], [41], [42], [35]. First, let {P n } n≥0 and {R n } n≥0 be sequences of monic polynomials with {u n } n≥0 and {w n } n≥0 their respective dual sequences, the following formula always holds:…”

Section: Finite-type Relationsmentioning

confidence: 99%

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On semi‐classical d‐orthogonal polynomials

Saib

2013

Mathematische Nachrichten

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In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation (ΦU)′=ΨU, where Φ and Ψ are d×d matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.

show abstract

{\{P_{n}\}}_{n \geq 0}

and

{\{R_{n}\}}_{n \geq 0}

be sequences of monic polynomials with

{\{u_{n}\}}_{n \geq 0}

and

{\{w_{n}\}}_{n \geq 0}

their respective dual sequences, the following formula always holds:

ϕ (x) R_{n} (x) = \sum_{ν = 0}^{n + t} λ_{n, v} P_{ν} (x), n \geq 0,

where

λ_{n, v} = 〈u_{v}, ϕ R_{n}〉,

0 \leq v \leq n + t,

n \geq 0 .

Definition When there is an integer

s \geq 0

such that we shall say that is a finite‐type relation between

{\{P_{n}\}}_{n \geq 0}

and

{\{R_{n}\}}_{n \geq 0}

, with respect to ϕ. Lemma For any sequence

{\{P_{n}\}}_{n \geq 0}

compatible with ϕ,

deg ϕ <...>

…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Section: Finite-type Relationsmentioning

confidence: 99%

On semi‐classical d‐orthogonal polynomials

Saib

2013

Mathematische Nachrichten

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“…Furthermore, there exist some extensions to the semi-classical situation [14]. The general theory of the structure relation for Askey-Wilson polynomials was presented by Koornwinder in [9].…”

Section: Introductionmentioning

confidence: 98%

Structure relations for monic orthogonal polynomials in two discrete variables

Rodal

Area

Godoy

2008

Journal of Mathematical Analysis and Applications

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In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector-matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.

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“…where P [1] n = P ′ n+1 /(n + 1), n ≥ 0 (see [8,18]). Recently, in [16], it was formulated the structure relation 2σ k=0 ξ n,k P n+σ−k (x) = σ+t k=0 ς n,k P [1] n+σ−k (x) , n ≥ max{t + 1, σ} , that, under certain conditions, establishes the semiclassical character of orthogonal polynomial sequences {P n }, and which generalizes the above mentioned second structure relation for classical orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…A theme of research in the theory of orthogonal polynomials is the characterization of orthogonal polynomial sequences satisfying a given structure relation. In this issue we refer the reader to [1,2], [4]- [8], [16]- [19] (see also [13], one of the first papers formulating inverse problems for orthogonal polynomials).…”

Section: Introductionmentioning

confidence: 99%

On the semi-classical character of orthogonal polynomials satisfying structure relations

Branquinho

Rebocho

2012

Journal of Difference Equations and Applications

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We prove the semiclassical character of some sequences of orthogonal polynomials, say {P n }, {R n }, related through relations of the following type: N k=0 ζ n,k R (α) n+i−k = M k=0 ξ n,k P n+j−k , where i, j, M, N, α are non-negative integers, ζ n,k , ξ n,k are complex numbers, and R (α) denotes the α-derivative of R. The case M = j = 0, α = 2, i = 2 is studied for a pair of orthogonal polynomials whose corresponding orthogonality measures are coherent. The relation s k=0 ξ n,k P n+s−k = s+2 k=0 ζ n,k P ′ n+s+1−k is shown to give a characterization for the semiclassical character of {P n }.

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Second structure relation for semiclassical orthogonal polynomials

Cited by 13 publications

References 13 publications

On semi‐classical d‐orthogonal polynomials

On semi‐classical d‐orthogonal polynomials

Structure relations for monic orthogonal polynomials in two discrete variables

On the semi-classical character of orthogonal polynomials satisfying structure relations

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