Polynomial solutions of hypergeometric type difference equations and their classification

Nikiforov, A. F.; Uvarov, V.B.

doi:10.1080/10652469308819023

Cited by 36 publications

(45 citation statements)

References 8 publications

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“…Relevant properties about orthogonality and the q-classical character of a sequence are summarised in §3.1, but we highlight the fact that an MOPS {P n } n 0 is q-classical if and only if there exist two polynomials (of degree at most 2) and (of degree 1) such that In other words, the elements of a q-classical MOPS are eigenfunctions of the second-order q-differential operator L q of q-Sturm-Liouville type (see Proposition 3.1 for more details). Regarding their importance in various fields of mathematics, there is a considerable bibliography on the subject -without any attempt for completion, we refer to [6], [10], [16], [17], [19], [20], [22], [23], [33]. In particular, they appear in the literature in the framework of discretisations of hypergeometric second order differential operators in q-lattices (see [20], [34]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

Loureiro

Zeng

2015

Mathematische Nachrichten

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ABSTRACT. We introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order qdifferential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q-version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q-version of the Jacobi-Stirling numbers given by Gelineau and the second author.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Regarding their importance in various fields of mathematics, there is a considerable bibliography on the subject -without any attempt for completion, we refer to [5,10,16,17,19,20,22,23,33]. In particular, they appear in the literature in the framework of discretisations of hypergeometric second order differential operators in q-lattices (see [20,34]).…”

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confidence: 99%

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

Loureiro

Zeng

2015

Mathematische Nachrichten

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“…In fact, comparing the di erence equation for the q-analog of the dual Hahn polynomials W (c) n (s; a; b) (2) with the recurrence relation for CGC's, we conclude that CGC's < J 1 M 1 J 2 M 2 jJM > q can be also expressed in terms of the q-dual Hahn polynomials as follows ( 1) (24) coincides with the right hand side of (19). Then, we can conclude that for the CGC's the following symmetry property holds (25) To conclude this Section we provide a table in which the corresponding properties of the Hahn q-polynomials h Finite di erence equation (2) …”

Section: X1 Introductionmentioning

confidence: 99%

The dual Hahnq-polynomials in the lattice and theq-algebras and

Álvarez-Nodarse

Smirnov

1996

J. Phys. A: Math. Gen.

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The dual q-Hahn polynomials in the non uniform lattice x(s) = s] q s + 1] q are obtained. The main data for these polynomials are calculated ( the square of the norm the coe cients of the three term recurrence relation, etc), as well as its representation as a q-hypergeometric series. The connection with the Clebsch-Gordan Coe cients of the Quantum Algebras SU q (2) and SU q (1; 1) is also given. x1 IntroductionIt is well known that the Lie Groups Representation Theory plays a very important role in the Quantum Theory and in the Special Function Theory. The group theory is an effective tool for the investigation of the properties of di erent special functions, moreover, it gives the possibility to unify various special functions systematically. In a very simple and clear way, on the basis of group representation theory concepts, the Special Function Theory was developed in the classical book of N.Ya.Vilenkin 1] and in the monography of N.Ya.Vilenkin and A.U.Klimyk 2], which have an encyclopedic character.In recent years, the development of the quantum inverse problem method 3] and the study of solutions of the Yang-Baxter equations 4] gave rise to the notion of quantum groups and algebras, which are, from the mathematical point of view, Hopf algebras 5]. They are of great importance for applications in quantum integrable systems, in quantum eld theory, and statistical physics (see 6] and references contained therein). They are attracting much attention in quantum physics, especially after the introduction of the q-deformed oscillator 7]-8]. Also they have been used for the description of the rotational and vibrational spectra of deformed nuclei 9]-11] and diatomic molecules 12]-14], etc. However to apply them it is necessary to have a well developed theory of their representations. In quantum physics, for instance, the knowledge of the Clebsch-Gordan coe cients (3j symbols ), Racah coe cients ( 6j symbols ) and 9j symbols 15] is crucial for applications because all 1 today 2 e-mail address: renato@dulcinea.uc3m.es, Fax (+341) 6249430.

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“…If we now use the power expansion of n (s), i.e., n (s) = 0 n x n (s)+ n (0) = 0 n q n x(s)+ n (0) and the TTRR (7) we obtain the rst structure relation (s) 5P n (s) q 5x(s) =S n P n+1 (s) q +T n P n (s) q +R n P n 1 (s) q ; (14) whereS n = n n] q q n n B n 0 n B n+1 ;…”

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confidence: 99%