q-Analogues of Some Multivariable Biorthogonal Polynomials

Gasper, George; Rahman, Mizan

doi:10.1007/0-387-24233-3_9

Cited by 3 publications

(4 citation statements)

References 9 publications

(9 reference statements)

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“…, a s+1 , η, N ) polynomials and of his second system of multivariable orthogonal Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases. Some q-extensions of Tratnik's [23] multivariable orthogonal Wilson polynomials and of his [21] multivariable biorthogonal generalization of the Wilson polynomials are given in our papers [11] and [10], respectively.…”

Section: Introductionmentioning

confidence: 99%

Some systems of multivariable orthogonal q-Racah polynomials

Gasper¹,

Rahman

2007

Ramanujan J

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Section: Introductionmentioning

confidence: 99%

Some systems of multivariable orthogonal q-Racah polynomials

Gasper¹,

Rahman

2007

Ramanujan J

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“…In 1991 Tratnik introduced some multivariable extensions of univariate orthogonal polynomials (see [51,52] and references therein). Moreover, q-analogues of these systems have been constructed by Gasper and Rahman [15,16,17], yielding systems of multivariable orthogonal Askey-Wilson polynomials and their special and limit cases. Bispectrality of multivariable Racah-Wilson and Askey-Wilson polynomials has been studied in [18] and [19], respectively.…”

Section: Introductionmentioning

confidence: 99%

Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions

Area

Atakishiyev

Godoy

et al. 2013

Applied Mathematics and Computation

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Orthogonal polynomial solutions of an admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type in two variables on q-linear lattices are analyzed. A q-Pearson's system for the orthogonality weight function, as well as for the difference derivatives of the solutions are presented, giving rise to a solution of the q-difference equation under study in terms of a Rodriguestype formula. The monic orthogonal polynomial solutions are treated in detail, giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy. Lewanowicz and Woźny [S. Lewanowicz, P. Woźny, J. Comput. Appl. Math. 233 (2010) 1554-1561] have recently introduced a (non-monic) bivariate extension of big q-Jacobi polynomials together with a partial q-difference equation of the hypergeometric type that governs them. This equation is analyzed in the last section: we provide two more orthogonal polynomial solutions, namely, a second non-monic solution from the Rodrigues' representation, and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series. Limit relations as q ↑ 1 for the partial q-difference equation and for the all three q-orthogonal polynomial solutions are also presented.

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“…Some q-extensions of Tratnik's [9] multivariable biorthogonal generalization of the Wilson polynomials are considered in this Proceedings [4]. q-Extensions of Tratnik's [11] system of multivariable orthogonal Racah polynomials and their special cases will be considered in a subsequent paper.…”

mentioning

confidence: 99%

“…In this paper we extend Tratnik's systems of multivariable Wilson polynomials to systems of multivariable Askey-Wilson polynomials and consider their special cases. Some q-extensions of Tratnik's [9] multivariable biorthogonal generalization of the Wilson polynomials are considered in this Proceedings [4]. q-Extensions of Tratnik's [11] system of multivariable orthogonal Racah polynomials and their special cases will be considered in a subsequent paper.…”

mentioning

confidence: 99%