Classical orthogonal polynomials: A functional approach

Marcellán, Francisco; Branquinho, A.; Petronilho, J.

doi:10.1007/bf00998681

Cited by 114 publications

(72 citation statements)

References 15 publications

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“…where φ and ψ are polynomials with deg(φ) Equation (5) is equivalent to the following two Equations [4,5] (which also characterize classical orthogonal polynomials), namely, the second-order differential Equation satisfied by each P n :…”

Section: Im(s(x + I Y) Dxmentioning

confidence: 99%

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

Foupouagnigni

Koepf

Kenfack-Nangho

et al. 2013

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Abstract:The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.

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Section: Im(s(x + I Y) Dxmentioning

confidence: 99%

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

Foupouagnigni

Koepf

Kenfack-Nangho

et al. 2013

Axioms

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“…Lately we encounter the growth of interest to the study of orthogonal polynomials relative an arbitrary (but still symmetric and nondegenerate) form, cf. [4,7,8] and references therein. In these approaches, however, the bilinear forms are introduced "by hands" and the differential or difference equations the orthogonal polynomials satisfy are of high degree.…”

Section: Introductionmentioning

confidence: 99%

Enveloping Superalgebra U(osp(1|2)) and Orthogonal Polynomials in Discrete Indeterminate

Сергеев¹

2001

JNMP

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Let A be an associative simple (central) superalgebra over C and L an invariant linear functional on it (trace). Let a → a t be an antiautomorphism of A such that (2))/m, where m is any maximal ideal of U (sl(2)), Leites and I have constructed orthogonal basis in A whose elements turned out to be, essentially, Chebyshev (Hahn) polynomials in one discrete variable. Here I take A = U (osp(1|2))/m for any maximal ideal m and apply a similar procedure. As a result we obtain either Hahn polynomials over C[τ ], where τ 2 ∈ C, or a particular case of Meixner polynomials, or -when A = Mat(n + 1|n) -dual Hahn polynomials of even degree, or their (hopefully, new) analogs of odd degree. Observe that the nondegenerate bilinear forms we consider for orthogonality are, as a rule, not sign definite.

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“…On the other hand, in [15], Marcellán and Pinzón Cortés considered a matrix interpretation of ðM; NÞ coherence of order m. They established a relation between the Jacobi matrices associated with ðM; NÞ coherent pairs of linear functionals of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the Sobolev inner product (11).…”

Section: A Matrix Characterization For the Coherence Of Orthogonal Pomentioning

confidence: 99%

“…In the classical case, i.e., s 0, we get that B is a ð0; 0Þ banded monic matrix, i.e., B is the identity matrix. Hence, fP n ðxÞg nP0 is classical if and only if BAA 1 AA 1 is a ð0; 2Þ banded monic matrix (result obtained in [20]) or, equivalently, fP n ðxÞg nP0 satisfies the following structure relation (proved in [11]) P n ðxÞ P ½1 n ðxÞ þ c n;n 1 P ½1 n 1 ðxÞ þ c n;n 2 P ½1 n 2 ðxÞ; n P 1:…”

Section: A Matrix Characterization For Semiclassical Polynomialsmentioning

confidence: 99%