Theory of recursive generation of systems of orthogonal polynomials: An illustrative example

Grosjean, C. C.

doi:10.1016/0377-0427(85)90026-3

Cited by 23 publications

(15 citation statements)

References 6 publications

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“…They were given in Refs. [3,10,15,16] for Jacobi, Hahn and Big-q-Jacobi (and also Little-qJacobi) respectively. and solving the linear system obtained with respect to the unknowns a n,k in terms of the coefficients of the polynomials A(s), B(s), C(s) and D(s).…”

Section: Applicationsmentioning

confidence: 99%

“…The four linearly independent solutions of the fourth-order divideddifference equations satisfied by the rth associated classical orthogonal polynomials are S 1 ðs; n; r; 0Þ ¼ rðsÞP r21 ðxðsÞP nþr ðxðsÞÞ; S 2 ðs; n; r; 0Þ ¼ rðsÞQ r21 ðxðsÞP nþr ðxðsÞÞ; S 3 ðs; n; r; 0Þ ¼ rðsÞP r21 ðxðsÞQ nþr ðxðsÞÞ; S 4 ðs; n; r; 0Þ ¼ rðsÞQ r21 ðxðsÞQ nþr ðxðsÞÞ: This is obtained by combining the previous theorem, (15), (16) and (107).…”

Section: Foupouagnigni 164mentioning

confidence: 99%

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On difference equations for orthogonal polynomials on nonuniform lattices¹

Foupouagnigni

2008

Journal of Difference Equations and Applications

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By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181-226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey -Wilson operator D q , following pioneering work by Magnus 1988, Associated Askey -Wilson polynomials as Laguerre-Hahn orthogonal polynomials, Lecture Notes in Mathematics (Berlin: Springer), vol. 1329, pp. 261-278, connecting D q and divided-difference operators. Finally, we propose a method to look for polynomial solutions of linear divided-difference equations with polynomial coefficients. Keywords: Classical orthogonal polynomials; Modifications of orthogonal polynomials; Divideddifference equations; quadratic and q-quadratic lattices; Functions of second kind Msc 2000: 33D45; 33C45; 33D20

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Section: Applicationsmentioning

confidence: 99%

Section: Foupouagnigni 164mentioning

confidence: 99%

On difference equations for orthogonal polynomials on nonuniform lattices¹

Foupouagnigni

2008

Journal of Difference Equations and Applications

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“…Nevai [14], Grosjean [8,9]). The following examples demonstrate that in some cases the results of Sections 1 and 2 can be useful for the identification of associated orthogonal polynomials and for the calculation of first return probabilities.…”

Section: Examplesmentioning

confidence: 99%

First return probabilities of birth and death chains and associated orthogonal polynomials

Dette

2000

Proc. Amer. Math. Soc.

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Abstract. For a birth and death chain on the nonnegative integers, integral representations for first return probabilities are derived. While the integral representations for ordinary transition probabilities given by Karlin and McGregor (1959) involve a system of random walk polynomials and the corresponding measure of orthogonality, the formulas for the first return probabilities are based on the corresponding systems of associated orthogonal polynomials. Moreover, while the moments of the measure corresponding to the random walk polynomials give the ordinary return probabilities to the origin, the moments of the measure corresponding to the associated polynomials give the first return probabilities to the origin.As a by-product we obtain a new characterization in terms of canonical moments for the measure of orthogonality corresponding to the first associated orthogonal polynomials. The results are illustrated by several examples.

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“…Askey and Wimp [2] determined the weight function and found an explicit formula for the generalized associated Laguerre and Hermite polynomials. A theory developed by Grosjean [7], [8], [9] permits us to deduce an explicit formula for the weight function u and a procedure for obtaining the basic interval with / meaning the Cauchy principal value integral.…”

Section: Ff])mentioning

confidence: 99%

“…Slightly modifying Grosjean's notation [8], [9], we define the polynomials associated with P<a-ß) by …”

Section: Ff])mentioning

confidence: 99%

Properties of the Polynomials Associated with the Jacobi Polynomials

Lewanowicz¹

1986

Mathematics of Computation

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Theory of recursive generation of systems of orthogonal polynomials: An illustrative example

Cited by 23 publications

References 6 publications

On difference equations for orthogonal polynomials on nonuniform lattices¹

On difference equations for orthogonal polynomials on nonuniform lattices¹

First return probabilities of birth and death chains and associated orthogonal polynomials

Properties of the Polynomials Associated with the Jacobi Polynomials

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Theory of recursive generation of systems of orthogonal polynomials: An illustrative example

Cited by 23 publications

References 6 publications

On difference equations for orthogonal polynomials on nonuniform lattices1

On difference equations for orthogonal polynomials on nonuniform lattices1

First return probabilities of birth and death chains and associated orthogonal polynomials

Properties of the Polynomials Associated with the Jacobi Polynomials

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On difference equations for orthogonal polynomials on nonuniform lattices¹

On difference equations for orthogonal polynomials on nonuniform lattices¹