Some remarks on a paper by L. Carlitz

Dominici, Diego

doi:10.1016/j.cam.2005.11.023

Cited by 5 publications

(5 citation statements)

References 43 publications

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“…We refer the reader to [16,21], for the case of non-classical Laguerre families; [17,3,2,1] for non-standard orthogonality concerning Jacobi polynomials; [4,10] for the case of Meixner polynomials with non-standard parameters; [7,8], for the case of symmetric Meixner-Pollaczek polynomials with parameters out of classical considerations; [11], for (not necessarily symmetric) generalized Meixner-Pollaczek polynomials with null parameter λ. .…”

Section: Non-standard Orthogonalitymentioning

confidence: 99%

Non-classical orthogonality relations for big and little q-Jacobi polynomials

Moreno

García−Caballero

2010

Journal of Approximation Theory

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Big q-Jacobi polynomials {P n (·; a, b, c; q)} ∞ n=0 are classically defined for 0 < a < q −1 , 0 < b < q −1 and c < 0. For the family of little q-Jacobi polynomials { p n (·; a, b|q)} ∞ n=0 , classical considerations restrict the parameters imposing 0 < a < q −1 and b < q −1 . In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditions for those cases for which Favard's theorem does not work. As a by-product, we obtain similar results for the families of big and little q-Laguerre polynomials.

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Section: Non-standard Orthogonalitymentioning

confidence: 99%

Non-classical orthogonality relations for big and little q-Jacobi polynomials

Moreno

García−Caballero

2010

Journal of Approximation Theory

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“…The orthogonality of some classical systems of polynomials, in the outstanding situation in which the three term recurrence relation breaks down (so the hypothesis of Favard's theorem does not hold), has been successfully developed in the last decade: see [21,24] for the Laguerre case, [22,4,2,1] for the Jacobi case, [3,12] for the Meixner case, [5,6] for the case of symmetric Meixner-Pollaczek polynomials, [13] for (not necessarily symmetric) Meixner-Pollaczek polynomials with null parameter and, finally, [12] for the classical families of polynomials which satisfy a discrete orthogonality with a finite number of masses (i.e., the Hahn, Racah, dual Hahn and Krawtchouk polynomials).…”

Section: Introductionmentioning

confidence: 99%

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

Moreno

García−Caballero

2010

Journal of Mathematical Analysis and Applications

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Keywords:Continuous q-ultraspherical polynomials Discrete q-ultraspherical polynomials Non-standard orthogonality Favard's theorem In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637-645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big qJacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey-Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.

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“…One such work that also develops a basic-type of characterization is [Di Bucchianico and Loeb 1994]. Other papers include [Al-Salam and Verma 1970;Di Bucchianico 1994;Di Nardo et al 2011;Dominici 2007;Hofbauer 1981;Popa 1997;1998;Shukla and Rapeli 2011]. In addition, a very large amount of work has been completed pertaining to the theory and applications of specific A-type 0 orthogonal sets, e.g., [Akleylek et al 2010;Chen et al 2011;Coffey 2011;Coulembier et al 2011;Dueñas and Marcellán 2011;Ferreira et al 2008;Hutník 2011;Khan et al 2011;Kuznetsov 2008;Miki et al 2011;Mouayn 2010;Sheffer 1941;Vignat 2011;Yalçinbaş et al 2011].…”

Section: Introductionmentioning

confidence: 99%

An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences

Galiffa

Riston

2015

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In 1939, Sheffer published "Some properties of polynomial sets of type zero", which has been regarded as an indispensable paper in the theory of orthogonal polynomials. Therein, Sheffer basically proved that every polynomial sequence can be classified as belonging to exactly one type. In addition to various interesting and important relations, Sheffer's most influential results pertained to completely characterizing all of the polynomial sequences of the most basic type, called A-type 0, and subsequently establishing which of these sets were also orthogonal. However, Sheffer's elegant analysis relied heavily on several characterization theorems. In this work, we show all of the Sheffer A-type 0 orthogonal polynomial sequences can be characterized by using only the generating function that defines this class and a monic three-term recurrence relation.

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Some remarks on a paper by L. Carlitz

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.

Cited by 5 publications

References 43 publications

Non-classical orthogonality relations for big and little q-Jacobi polynomials

Non-classical orthogonality relations for big and little q-Jacobi polynomials

New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences

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