Variations around classical orthogonal polynomials. Connected problems

Maroni, P.

doi:10.1016/0377-0427(93)90319-7

Cited by 111 publications

(81 citation statements)

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“…D-semiclassical orthogonal polynomials were introduced in a seminal paper by J. A. Shohat [22] and extensively studied by P. Maroni and coworkers in the last decade [13][14][15][16][17][18][19]. Furthermore, the present contribution is a natural continuation of a previous work [9] by me and P. Maroni on q-classical orthogonal polynomials.…”

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confidence: 61%

AN INTRODUCTION TO THE H<sub>q</sub>-SEMICLASSICAL ORTHOGONAL POLYNOMIALS

Khériji¹

2003

Methods and Applications of Analysis

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Abstract. Orthogonal polynomials associated with Hq− semiclassical linear form will be studied as a generalization of the Hq−classical linear forms. The concept of class and a criterion for determining it will be given. The q-difference equation that the corresponding formal Stieltjes series satisfies is obtained. Also, the structure relation as well as the second order linear q-difference equation are obtained. Some examples of Hq−semiclassical of class 1 were highlighted.Introduction. The aim of this paper is to present the analysis and characterization of the q-analogues of D-semiclassical orthogonal polynomials. D-semiclassical orthogonal polynomials were introduced in a seminal paper by J. A. Shohat [22] and extensively studied by P. Maroni and coworkers in the last decade [13][14][15][16][17][18][19]. Furthermore, the present contribution is a natural continuation of a previous work [9] by me and P. Maroni on q-classical orthogonal polynomials.In the literature, the extension of classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) can be done in the q-case from three basic approaches (see [2] for a comparative analysis).The first one is related with the so called Askey Tableau, where all the classical families appear in a limiting process from the top of Askey-Wilson polynomials (see [10]).The second one concerns the hypergeometric character of classical orthogonal polynomials , i.e. as solutions of a second order linear differential equation with polynomial coefficients, the so called Nikiforov-Uvarov approach (see [21]).The third one is based in the Pearson equation which satisfies the symmetric factor for the above differential equation. This idea appears in several papers but the basic theory was developed by P. Maroni.The structure of this paper is as follows: The first section contains material of preliminary and introductory character. Instead of the derivative operator, we use the q-operator H q introduced by Hahn [7]. In particular, we define a H q −semiclassical linear form u from a functional equation which is the q−difference distributional Pearson one. The second section deals with so-called class of H q −semiclassical linear forms. A criterion for determining it is given. In the third section, we establish the different characterizations of H q −semiclassical linear forms. We can characterize a H q −semiclassical linear form through the fact that its Stieltjes function satisfies a first order linear q−difference equation with polynomial coefficients. A second characterization is the so-called structure relation that the polynomials {P n } n≥0 orthogonal with respect to u satisfy. It is deduced from theory of finite-type relations between polynomial sequences [19]. A third characterization is the second order linear q−difference equation satisfied by P n+1 , n ≥ 0. Lastly, in section 4 we construct some examples of H q −semiclassical linear forms of class 1 by taking into account a method studied by P. Maroni in [15] for the D-case ( see paragraph 5.1 below ) and by using s...

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confidence: 61%

AN INTRODUCTION TO THE H<sub>q</sub>-SEMICLASSICAL ORTHOGONAL POLYNOMIALS

Khériji¹

2003

Methods and Applications of Analysis

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“…From [11] we know that these polynomials have some properties related to the classical orthogonal Jacobi polynomials [9,12], and their generating function is…”

Section: Examples and Numerical Resultsmentioning

confidence: 99%

“…From those coefcients, we compute some FPA, and we obtain again good approximation results. Finally, in order to compare FPA of dimension 1 and 2, we compute the rst coecients of the developments of the Airy function as a 2-OS of Laguerre type and a 1-OS of Laguerre [9,12], then we compute some 2-FPA and 1-FPA respectively, and we compare them. The results show that FPA of dimension 2 are better than the ones of dimension 1.…”

Section: Introductionmentioning

confidence: 99%

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

Matos

Rocha

2005

Applied Numerical Mathematics

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“…It is well known that the structure relation (1.1)-(1.2) of a regular monic orthogonal PS (MOPS) becomes the following second order recurrence relation [1,7], since χ n,ν = 0, 0 ≤ ν < n, n ≥ 0, and recalling that γ n+1 = χ n,n = 0, n ≥ 0,…”

Section: Basic Notions and Fundamental Resultsmentioning

confidence: 99%

Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases

Mesquita¹,

Rocha²

2012

OpMath

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Abstract. We deal with a symbolic approach to the cubic decomposition (CD) of polynomial sequences -presented in a previous article referenced herein -which allows us to compute explicitly the first elements of the nine component sequences of a CD. Properties are investigated and several experimental results are discussed, related to the CD of some widely known orthogonal sequences. Results concerning the symmetric character of the component sequences are established.

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Variations around classical orthogonal polynomials. Connected problems

Cited by 111 publications

References 6 publications

AN INTRODUCTION TO THE H<sub>q</sub>-SEMICLASSICAL ORTHOGONAL POLYNOMIALS

AN INTRODUCTION TO THE H<sub>q</sub>-SEMICLASSICAL ORTHOGONAL POLYNOMIALS

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

Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases

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