Journal ArticleDOI
Polynomial solutions of hypergeometric type difference equations and their classification
TLDR
For the most general case of hypergeometric type difference equations on uniform and nonuniform lattices, the explicit expression of polynomials solutions in terms of generalized q-hypergeometric series is obtained from the difference analog of the Rodrigues formula as discussed by the authors.Abstract:
For the most general case of hypergeometric type difference equations on uniform and nonuniform lattices the explicit expression of polynomials solutions in terms of generalized q-hypergeometric series is obtained from the difference analog of the Rodrigues formula. From this expression, the formulas for all particular cases are derived by an appropriate choice of parameters and by taking various limits. Basic hypergeometric series are also used [14]. The consideration of these polynomials gives us the classification of corresponding q-polynomials in accordance with the values of parameters q, μ of the lattice , and with the number of zeroes of the function σ(s) that is the coefficient before the senior difference derivative in the equation of hypergeometric type. All earlier introduced q-polynomials are included in our scheme of classification. This review is an expanded version of the report which was made by one of the authors on the VII simposium sobre polinomios ortogonales y aplicationes (23-27 Sept...read more
Citations
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The history of q-calculus and a new method
TL;DR: In this paper, the authors considered the problem of additive number theory for the Riemann surface and showed that the solution of the problem can be expressed as a polynomial series.
Journal ArticleDOI
On characterizations of classical polynomials
TL;DR: In this article, a unified study of the classical discrete polynomials and q-polynomials of the q-Hahn tableau is presented by using the difference calculus on linear-type lattices.
Journal ArticleDOI
On the q -polynomials: a distributed study
TL;DR: From the distributional q-Pearson equation, many of the properties of classical discrete q-polynomials are deduced such as the three-term recurrence relations, structure relations, etc.
Journal ArticleDOI
q -Classical polynomials and the q -Askey and Nikiforov-Uvarov tableaus
TL;DR: In this paper, the q-polynomials are compared with the Nikiforov-Uvarov tableau and the Askey-Askey tableau, and a new family of q polynomials is introduced.
Journal ArticleDOI
Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials
TL;DR: In this article, the authors studied the connection problem between discrete hypergeometric polynomials and the product case with m = 0 and its complete solution for all the classical discrete orthogonal hypergeometrical polynomial (CDOH) polynomorphisms is given.
References
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Book
Basic Hypergeometric Series
George Gasper,Mizan Rahman +1 more
TL;DR: In this article, the Askey-Wilson q-beta integral and some associated formulas were used to generate bilinear generating functions for basic orthogonal polynomials.
BookDOI
Special functions of mathematical physics : a unified introduction with applications
TL;DR: The theory of classical or thogonal polynomials of a discrete variable on both uniform and non-uniform lattices has been given a coherent presentation, together with its various applications in physics as discussed by the authors.
Book
Generalized Hypergeometric Series
TL;DR: Koornwinder as discussed by the authors gave identitity (2.5) with N = 0 and formulas (5.3), 5.3, and 5.4) substituted.
Book
Some Basic Hypergeometric Orthogonal Polynomials That Generalize Jacobi Polynomials
Richard Askey,James A. Wilson +1 more
Book
Classical Orthogonal Polynomials of a Discrete Variable
TL;DR: In this article, the orthogonality relation (2.0.1) is reduced to 2.0, where w(x) is a function of jumps, i.e. the piecewise constant function with jumps ϱ i at the points x = x i.