Classical Orthogonal Polynomials of a Discrete Variable

Nikiforov, A. F.; Uvarov, V.B.; Suslov, Sergeĭ K.

doi:10.1007/978-3-642-74748-9

Cited by 584 publications

(735 citation statements)

References 6 publications

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“…we denote the q-factorial which satis es the relation x + 1] q ! = x + 1] q x] q !, and coincides with the~ q (x) function introduced by Nikiforov et al ( 24] (11) In order to nd the representation of these polynomials in terms of q-Hypergeometric );~ n 1 (s) = n (s ) q : (14) The formula (14) will be called the rst di erentiation formula for the polynomials W ; s 0 = s 1 2 , we nd n+1 (s 1 2 ; a 0 ; b 0 ; c 0 ) = q 2n+a+c b+ 1 4 n (s; a; b; c): (15) Then, from the Rodrigues Formula (5) P n+1 (s and c 0 = c 1 2 . x4 Clebsch-Gordan coe cients for the q-algebra SU q (2) and the dual Hahn q-polynomials.…”

Section: X1 Introductionsupporting

confidence: 72%

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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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Section: X1 Introductionsupporting

confidence: 72%

“…It is known ( 24] and 25]) that for some special kind of lattices, solutions of (2) are orthogonal polynomials of a discrete variable, in other words, they satisfy the orthogonality relation x(s)P n (s) = n P n+1 (s) + n P n (s) + n P n 1 (s); P 1 (s) = 0; P 0 (s) = 1: (4)…”

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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“…It should be noted that, because of relation (8), R +2R = m and consequently the parameters 1 and M may be written in the form…”

Section: Determining the Discrete Density Of Zerosmentioning

confidence: 99%

“…To go further in the analysis of the N dependence of µ (N ) m one has to analyse expression (65) which defines 1 . A simple study allows us to distinguish the following three situations:…”

Section: Determining the Discrete Density Of Zerosmentioning

confidence: 99%

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The distribution of zeros of generalq-polynomials

Álvarez-Nodarse

Buendı́a

Dehesa

1997

J. Phys. A: Math. Gen.

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Abstract.A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation. This result is applied to specific classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey and Wilson, Al Salam-Carlitz and the celebrated little and big q-Jacobi.

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Advances in Robust Parameter Design: From Taguchi's Inner‐Outer Arrays to Combined Arrays

Evangelaras

Koukouvinos

Koutras

2005

Encyclopedia of Statistical Sciences

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Robust parameter design, originally proposed by Taguchi, is an off‐line technique for maintaining variance reduction and improving quality. According to Taguchi, product variation from the desirable target is affected by two types of factors: control factors and noise factors. The basic idea in robust parameter design is to identify the levels of the control factors so that the product's quality characteristic becomes insensitive to changes in the levels of the noise factors. As a consequence, the effect of uncontrollable variations on the response will be reduced. Taguchi suggested the use of inner‐outer arrays with the control factors being assigned to an inner array and the noise factors assigned to an outer array. After the collection of the data, a performance measure called signal‐to‐noise ratio is calculated and analyzed in order to identify the optimal levels of the control factors. After the introduction of Taguchi's philosophy in the field of experimental design and data analysis, several new designs and techniques have been explored and adopted in this field. This chapter overviews well established approaches and analysis techniques that have been exploited to the robust parameter design field.

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Classical Orthogonal Polynomials of a Discrete Variable

Cited by 584 publications

References 6 publications

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

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

The distribution of zeros of generalq-polynomials

Advances in Robust Parameter Design: From Taguchi's Inner‐Outer Arrays to Combined Arrays

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