International Journal of Mathematics and Mathematical Sciences, 1999
Let{Pn}n≥0be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0andω1(s... more Let{Pn}n≥0be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0andω1(see Definition 1.1). Now, let{Qn}n≥0be the sequence of polynomials defined byQn:=(n+1)−1P′n+1,n≥0. When{Qn}n≥0is, also, 2-orthogonal,{Pn}n≥0is called classical (in the sense of having the Hahn property). In this case, both{Pn}n≥0and{Qn}n≥0satisfy 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 differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionalsω0andω1and obtain their weight functions which satisfy a second-order differential equation. From all these properties, we show that the resulting polynomials are an ...
Remember that a d-orthogonal polynomial sequence (d-OPS) is defined by systems of orthogonality r... more Remember that a d-orthogonal polynomial sequence (d-OPS) is defined by systems of orthogonality relations or, equivalently, a polynomial sequence satisfying a (d+1)-order recurrence relation (see Definition 1.1). In this paper we consider only the problem (1). The resulting ...
Abstract. Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear function... more Abstract. Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0 andω1 (see Definition 1.1). Now, let {Qn}n≥0 be the sequence of polynomi-als defined by Qn: = (n+1)−1P ′n+1, n ≥ 0. When {Qn}n≥0 is, also, 2-orthogonal, {Pn}n≥0 is called “classical ” (in the sense of having the Hahn property). In this case, both {Pn}n≥0 and {Qn}n≥0 satisfy a third-order recurrence relation (see below). Working on the recur-rence coefficients, under certain assumptions and well-chosen parameters, a classical fam-ily of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionals ω0 and ω1 and obtain their weight functions which satisfy a second-order differential equation. From all these prop-erties, we sho...
The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they ar... more The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
ABSTRACT The classical 2-orthogonal polynomials share the so-called Hahn property, this means tha... more ABSTRACT The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they ar... more The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 2000
In this Note, we present an extension of Abdul-Halim and Al-Salam's result [1] in the contex... more In this Note, we present an extension of Abdul-Halim and Al-Salam's result [1] in the context of d -orthogonality. The resulting polynomials are analogous to the classical Laguerre polynomials. We provide some of their properties.
ABSTRACT We give a characterization of "classical" d-orthogonal polynomials thr... more ABSTRACT We give a characterization of "classical" d-orthogonal polynomials through a vectorial functional equation. A sequence of monic polynomials {Bn}n ≥ 0 is called d-simultaneous orthogonal or simply d-orthogonal if it fulfils the following d + 1-st order recurrence relation: [formula] with the initial conditions [formula] Denoting by {n}n ≥ 0 the dual sequence of {Bn}n ≥ 0, defined by 〈n, Bm〉 = δn, m, n, m ≥ 0, then the sequence {Bn}n ≥ 0 is d-orthogonal if and only if [formula] for any integer α with 0 ≤ α ≤ d − 1. Now, the d-orthogonal sequence {Bn}n ≥ 0 is called "classical" if it satisfies the Hahn′s property, that is, the sequence {Qn}n ≥ 0 is also d-orthogonal where Qn(x) = (n + 1)− 1 B′n + 1(x), n ≥ 0 is the monic derivative. If Λ denotes the vector t(0, 1, ..., d − 1), the main result is the following: the d-orthogonal sequence {Bn}n ≥ 0 is "classical" if and only if, there exist two d × d polynomial matrices Ψ = (ψν, μ), Φ = (φν, μ), deg ψν, μ ≤ 1, deg φν, μ ≤ 2 such that Ψ Λ + D(Ψ Λ) = 0 with conditions about regularity (see below). Moreover, some examples are given.
Información del artículo On Two-Orthogonal Polynomials Related to the Bateman's J"uvn&q... more Información del artículo On Two-Orthogonal Polynomials Related to the Bateman's J"uvn"_Function.
International Journal of Mathematics and Mathematical Sciences, 1999
Let{Pn}n≥0be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0andω1(s... more Let{Pn}n≥0be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0andω1(see Definition 1.1). Now, let{Qn}n≥0be the sequence of polynomials defined byQn:=(n+1)−1P′n+1,n≥0. When{Qn}n≥0is, also, 2-orthogonal,{Pn}n≥0is called classical (in the sense of having the Hahn property). In this case, both{Pn}n≥0and{Qn}n≥0satisfy 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 differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionalsω0andω1and obtain their weight functions which satisfy a second-order differential equation. From all these properties, we show that the resulting polynomials are an ...
Remember that a d-orthogonal polynomial sequence (d-OPS) is defined by systems of orthogonality r... more Remember that a d-orthogonal polynomial sequence (d-OPS) is defined by systems of orthogonality relations or, equivalently, a polynomial sequence satisfying a (d+1)-order recurrence relation (see Definition 1.1). In this paper we consider only the problem (1). The resulting ...
Abstract. Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear function... more Abstract. Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0 andω1 (see Definition 1.1). Now, let {Qn}n≥0 be the sequence of polynomi-als defined by Qn: = (n+1)−1P ′n+1, n ≥ 0. When {Qn}n≥0 is, also, 2-orthogonal, {Pn}n≥0 is called “classical ” (in the sense of having the Hahn property). In this case, both {Pn}n≥0 and {Qn}n≥0 satisfy a third-order recurrence relation (see below). Working on the recur-rence coefficients, under certain assumptions and well-chosen parameters, a classical fam-ily of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionals ω0 and ω1 and obtain their weight functions which satisfy a second-order differential equation. From all these prop-erties, we sho...
The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they ar... more The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
ABSTRACT The classical 2-orthogonal polynomials share the so-called Hahn property, this means tha... more ABSTRACT The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they ar... more The classical 2-orthogonal polynomials share the so-called Hahn property, this means that they are 2-orthogonal polynomials whose the sequences of their derivatives are also 2-orthogonal polynomials. Based only on this property, a new class of classical 2-orthogonal polynomials is obtained as particular solution of the non-linear system governing the coefficients involved in the recurrence relation fulfilled by these polynomials. A differential-recurrence relation as well as a third-order differential equation satisfied by the resulting polynomials are given. Many interesting subcases are highlighted and explicitly presented with special reference to some connected results that exist in the literature. The integral representations of their associated linear functionals will be exhaustively discussed in a forthcoming publication.
Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 2000
In this Note, we present an extension of Abdul-Halim and Al-Salam's result [1] in the contex... more In this Note, we present an extension of Abdul-Halim and Al-Salam's result [1] in the context of d -orthogonality. The resulting polynomials are analogous to the classical Laguerre polynomials. We provide some of their properties.
ABSTRACT We give a characterization of "classical" d-orthogonal polynomials thr... more ABSTRACT We give a characterization of "classical" d-orthogonal polynomials through a vectorial functional equation. A sequence of monic polynomials {Bn}n ≥ 0 is called d-simultaneous orthogonal or simply d-orthogonal if it fulfils the following d + 1-st order recurrence relation: [formula] with the initial conditions [formula] Denoting by {n}n ≥ 0 the dual sequence of {Bn}n ≥ 0, defined by 〈n, Bm〉 = δn, m, n, m ≥ 0, then the sequence {Bn}n ≥ 0 is d-orthogonal if and only if [formula] for any integer α with 0 ≤ α ≤ d − 1. Now, the d-orthogonal sequence {Bn}n ≥ 0 is called "classical" if it satisfies the Hahn′s property, that is, the sequence {Qn}n ≥ 0 is also d-orthogonal where Qn(x) = (n + 1)− 1 B′n + 1(x), n ≥ 0 is the monic derivative. If Λ denotes the vector t(0, 1, ..., d − 1), the main result is the following: the d-orthogonal sequence {Bn}n ≥ 0 is "classical" if and only if, there exist two d × d polynomial matrices Ψ = (ψν, μ), Φ = (φν, μ), deg ψν, μ ≤ 1, deg φν, μ ≤ 2 such that Ψ Λ + D(Ψ Λ) = 0 with conditions about regularity (see below). Moreover, some examples are given.
Información del artículo On Two-Orthogonal Polynomials Related to the Bateman's J"uvn&q... more Información del artículo On Two-Orthogonal Polynomials Related to the Bateman's J"uvn"_Function.
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