To prove that **the** above expression is a polynomial **of** degree p + 1 we substitute (1.10), p (x) = a p x p + · · · , **and** equate **the** coefﬁcients **of** x p+ 1 . This gives a p {M + A( 2 n − p − 2 )} = 0, due to **the** regularity condition (see Theorem 1.6) **and** **the** fact that a p = 0 since **the** polynomial p has degree equal to p. This prove (2.2). Now putting k = n − 1, **the** result follows.

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lattices. On **the** other hand, using **the** **q**-analog **of** **the** quantum theory **of** angular momentum [20–23] we can obtain several results for **the** **q**-**polynomials**, some **of** which are nontrivial from **the** viewpoint **of** **the** theory **of** **orthogonal** **polynomials** (see, e.g., **the** nice surveys [24, 25]). In fact, in this paper we present a detailed study **of** some **q**-analogs **of** **the** Racah **polynomials** u α,β n (x(s), a, b) **q** **and** u e

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as a modication **of** **the** rst ones troughtout **the** addition **of** two mass points. All **the** formulas for **the** **classical** Hahn **polynomials** can be found in a lot **of** books ( see for instance **the** excellent monograph **Orthogonal** **Polynomials** in **Discrete** Variables by A.F. Nikiforov, S. K. Suslov, V. B. Uvarov [16], Chapter 2.)

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Special important examples appear when is a **classical** linear functional **and** **the** mass points are located at **the** ends **of** **the** interval or **orthogonal**- ity. In this case we have studied **the** corresponding sequences **of** **orthogonal** **polynomials** in several papers: For **the** Laguerre linear functional see [2, 3], for **the** Bessel linear functional see [6], for **the** Jacobi linear functional see [7] **and** for **the** Hermite case see [1]. In particular **the** quasi-definiteness **of** e , relative asymptotics, **the** representation as a hypergeometric function, **and** **the** location **of** their zeros have been obtained.

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We have used a method [12, 16, 17] which is based only on **the** three-term recurrence relation satised by **the** involved **polynomials**. This method, which will be described in Section 2, is **of** general vality since no peculiar constraints are imposed upon **the** coecients **of** **the** recurrence relation. It was found in a context **of** tridiagonal matrices [6, 13, 14, 15] **and** it has been already used for **the** study **of** **the** distribution **of** zeros **of** **q**-**polynomials** [1, 11, 16]. Some **of** **the** results found here have been previously obtained by other means **and** are dispersely published, what will be mentioned in **the** appropiate place; they are included here for completeness, for illustrating **the** goodness **of** our procedure or because they are not accessible for **the** general reader [16].

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It 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 ef- fective tool for **the** investigation **of** **the** properties **of** dierent 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.

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In **the** last decade an increasing interest on **the** so called **q**-**orthogonal** **polynomials** (or basic **orthogonal** **polynomials**) is observed ( for a review see [1], [2] **and** [3]). **The** reason is not only **of** purely intrinsic nature but also because **of** **the** so many applications in several areas **of** Math- ematics ( e.g., continued fractions, eulerian series, theta functions, elliptic functions,...; see for instance [4] **and** [5]) **and** Physics ( e.g., angular momentum [6] **and** [7] **and** its **q**-analog [8]-[11], **q**-Shrodinger equation [12] **and** **q**-harmonic oscillators [13]-[19]). Moreover, it is well known **the** connection between **the** representation theory **of** quantum algebras (Clebsch-Gordan coecients, 3j **and** 6j symbols) **and** **the** **q**-**orthogonal** **polynomials**, (see [20], [21] (Vol. III), [22], [23], [24] ), **and** **the** important role that these **q**-algebras play in physical applications (see for instance [26]-[31] **and** references therein).

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Remark 2.17. Both **the** distributional equation (2.11) **and** **the** Sturm–Liouville equation (2.12), char- acterize a **q**-**classical** functional **and** its corresponding OPS by means **of** **and** . **The** 1rst one is a di<erential equation **of** 1rst order which is easier to use than **the** second one which is **of** sec- ond order. Nevertheless, **the** Sturm–Liouville equation has **the** advantage that is an equation in **the** space **of** **polynomials** **and** combined with **the** TTRR (1.1) gives an alternative method to prove **the** quasi-de1niteness **of** **the** functional instead **of** **the** analysis **of** **the** Hankel determinants (see Theorem 1.6) as we already pointed out in **the** previous remark.

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We underline that, in a different context (cf. [2, Theorem 2]), the characterization of solutions of an integrable system was established in terms of the derivative of the polynomials a[r]

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A matrix versión of this theorem, (see [5,pp. 68]), and Zhedanov has constructed, using the symmetrized Al-Salam-Carlitz polynomials, examples of orthogonal polynomials for a discrete m[r]

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Special cases **of** quasi-definite linear functionals are **the** **classical** ones (those **of** Ja- cobi, Laguerre, Hermite, **and** Bessel). In **the** last years perturbations **of** **the** func- tional via **the** addition **of** Dirac delta functions —**the** so-called Krall-type orthog- onal **polynomials**— have been extensively studied (see e.g. [6, 7, 16, 20, 21, 22, 26] **and** references therein), i.e., **the** linear functional

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k = b − s − 1. Here we do this for **the** **classical** **and** alternative Hahn **and** Racah **polynomials** as well as for their **q**-analogs. Also we establish **the** connection between **classical** **and** alternative families. This allows us to obtain new expressions for **the** Clerbsch–Gordan **and** Racah coefficients **of** **the** quantum algebra U **q** (su(2)) in terms **of** various Hahn **and** Racah **q**-**polynomials**.

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nality class is called semi-**classical** **and** is very large [11], [7] . **The** **classical** (con- tinuous) family: Jacobi, Bessel, Laguerre, Hermite (see for instance [12]) **and** **the** **classical**(**discrete**) family: Hahn , Kravchuk, Meixner, Charlier (see for instance [13]) are **of** course included in **the** semi-**classical** class. When **the** orthogonality measure is dened by a weight ( x ), **the** semi-**classical** class covers all weights solution **of** a linear rst order dierential (or dierence) equation with polynomial coecients.

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Uvarov, Polynomial Solutions of hypergeometri type dierene Equations. and their lassiation[r]

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