The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

Grosjean, C. C.

doi:10.1016/0377-0427(86)90001-4

Cited by 46 publications

(28 citation statements)

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“…This relation was already found by Grosjean [9] using a special structure of the corresponding weight function and by Wimp [22] [formula (40)] as a special case of a representation for the c-th associated Jacobi polynomial in terms of generalized hypergeometric functions.…”

Section: Examplessupporting

confidence: 59%

“…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%

“…Note that Theorem 2.1 gives a new characterization of the measure of orthogonality corresponding to the first associated orthogonal polynomials, which is different from the representations commonly used in the literature (see e.g. Sherman [15] or Grosjean [9]). For example, it is shown in Van Assche [19] that the Stieltjes transform of the orthogonality measure corresponding to the first associated orthogonal polynomials is given by the equation (2.6) and ψ (1) could be determined from ψ by the Stieltjes inversion formula.…”

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confidence: 95%

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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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Section: Examplessupporting

confidence: 59%

Section: Examplesmentioning

confidence: 99%

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

Dette

2000

Proc. Amer. Math. Soc.

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“…Here, G r21 and g 0 are given respectively by (16) and (112). Then, sinceP n ðxðsÞÞ and S 2 (s;n,r,k) are both solutions of (120), it turns out from the previous equation that S 3 ðs; n; r; kÞ is also solution of (120).…”

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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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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“…So, we have[18,34] G ( ) n (x) = c n P ( ,−1− ) n (x), −1 < < 0 and g ( ) n (x) = e n P ( ,1− ) n (x), −1 < < 2, for the Grosjean polynomials of first and second kind, respectively, with the values c n = 2 n 2n − 1 n (G) n ( )] 2 = 2 2n−1 2 (n) 2 (2n) (n + + 1) (n − ).…”

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Parameter-based Fisher's information of orthogonal polynomials

Dehesa

Olmos

Yáñez

2008

Journal of Computational and Applied Mathematics

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The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

Cited by 46 publications

References 4 publications

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

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

On difference equations for orthogonal polynomials on nonuniform lattices¹

Parameter-based Fisher's information of orthogonal polynomials

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The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

Cited by 46 publications

References 4 publications

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

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

On difference equations for orthogonal polynomials on nonuniform lattices1

Parameter-based Fisher's information of orthogonal polynomials

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Product

Resources

About

On difference equations for orthogonal polynomials on nonuniform lattices¹