Les Polynômes Orthogonaux „classiques“ De Dimension Deux

Douak, Khalfa; Maroni, P.

doi:10.1524/anly.1992.12.12.71

Cited by 77 publications

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“…Actually, there is, in the literature (see for example [3,4,5,6,10,11]), many information about orthogonal polynomials of dimension d, especially for the case d = 2. These polynomials can be used most likely orthogonal polynomials.…”

Section: Resultsmentioning

confidence: 99%

Frobenius–Padé approximants for d-orthogonal series: Theory and computational aspects

Matos

Rocha

2005

Applied Numerical Mathematics

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Section: Resultsmentioning

confidence: 99%

Frobenius–Padé approximants for d-orthogonal series: Theory and computational aspects

Matos

Rocha

2005

Applied Numerical Mathematics

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“…Then P n = Q n ,n = 0, 1, 2,... and {P n } n≥0 is, at the same time, 2-orthogonal and Appell sequence [5,6]. In this case, the polynomials obtained are the analogous of the classical orthogonal polynomials of Hermite.…”

Section: The Case (A)mentioning

confidence: 95%

“…Otherwise, when the 2-OPS {P n } n≥0 is "classical", it was obtained in [6] that the recurrence coefficients {β n }, {β n }, {γ ν n }, and {γ ν n }(ν = 0, 1) satisfy the following non-linear system which is valid for n ≥ 1 :…”

Section: Introductionmentioning

confidence: 99%

“…However, if we impose symmetry [6] or are interested in particular conditions [5,9,15,8], some solutions have been obtained. For instance, under the assumption (δ n = 0,n ≥ 0, then β n = β n = const.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, under the assumption (δ n = 0,n ≥ 0, then β n = β n = const. ), the system was solved in [6] and a class of "classical" 2-orthogonal polynomials were obtained. In this paper, we give another particular solution when δ n = 0,n ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

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On 2‐orthogonal polynomials of Laguerre type

Douak

1999

International Journal of Mathematics and Mathematical Sciences

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Abstract. Let {P n } n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionals ω 0 and ω 1 (see Definition 1.1). Now, let {Q n } n≥0 be the sequence of polynomials defined by Q n := (n + 1) −1 P n+1 , n ≥ 0. When {Q n } n≥0 is, also, 2-orthogonal, {P n } n≥0 is called "classical" (in the sense of having the Hahn property). In this case, both {P n } n≥0 and {Q n } n≥0 satisfy a third-order recurrence relation (see below). Working on the recurrence coefficients, under certain assumptions and well-chosen parameters, a classical family of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differentialrecurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionals ω 0 and ω 1 and obtain their weight functions which satisfy a second-order differential equation. From all these properties, we show that the resulting polynomials are an extention of the classical Laguerre's polynomials and establish a connection between the two kinds of polynomials. They are defined by the generating functionTheir corresponding monic polynomials L (α) n n≥0 are defined byLn , n ≥ 0 and satisfy the second-order recurrence relation 30KHALFA DOUAK as well as the following relations:Recently, within the framework of the d-orthogonality of polynomials or polynomials of simultaneous orthogonality studied in [12,11,16] which does not really have the same orthogonality relations but are considered to be orthogonal relative to positive measures, new kinds of d-orthogonal polynomials have been the subject of various investigations [1,3,5,9,15]. In particular, those having some properties that are analogous to the classical orthogonal polynomials.In this paper, when d = 2, under special conditions and well-chosen parameters, we give a family of 2-orthogonal "classical" polynomials which are a natural extension of the classical Laguerre polynomials. These polynomials have some properties analogous to those satisfied by the classical Laguerre polynomials. Their recurrence coefficients and generating function are explicitly determined, a differential-recurrence relation and a third-order differential equation are obtained. We denote these polynomials by P n (·; α), where α is an arbitrary parameter. They are called the 2-orthogonal polynomials of Laguerre type related to the two linear functionals ω 0 ,ω 1 , where ω 0 satisfies a second-order differential (distributional) equation and ω 1 is given in terms of ω 0 and ω 0 (see equations (4.13), (4.14)). Finally, one of the problems is to determine integral representations of both functionals ω 0 and ω 1 . Indeed, applying the method explained in [8], if we denote by ᐃ 0 (resp., ᐃ 1 ) the weight function representing the functional ω 0 (resp., ω 1 ), we obtain that when α > −1, ᐃ 0 (x) = e −1 (x)I * α (x) on the interval 0 ≤ x < +∞, with (x) = x α e −x being the weight function related t...

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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.

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Les Polynômes Orthogonaux „classiques“ De Dimension Deux

Cited by 77 publications

References 0 publications

Frobenius–Padé approximants for d-orthogonal series: Theory and computational aspects

Frobenius–Padé approximants for d-orthogonal series: Theory and computational aspects

On 2‐orthogonal polynomials of Laguerre type

On semi‐classical d‐orthogonal polynomials

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