In this case we have 12 families of classical **q**-**polynomials**, (see [2,10]). We will take two representatives examples corresponding to the big **q**-Jacobi **polynomials** P n (x, a, b, c; **q**) and the little **q**-Jacobi **polynomials** p n (x; a, b|**q**) . The main data of such **polynomials** can be found in [2,5]. The results are given in Table 2. The other ten cases can be obtained in an analogous way or by taking appropriate limits (see e.g. [2,5]).

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Proofs and detailed discussion of these theorems are contained in Sections 4 and 5 respectively. The utmost eort has been concentrated on searching for an appropiate asymptotic density of zeros to obtain as much information as possible about the asymptotic distribution of zeros of the new **polynomials**. Finally, Section 6 contains application of theorems 1, 2, 3 and 4 formulated in section 3 to several known families of **q**-**polynomials**.

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In this paper we wish to apply the technique of [7] to the study of those families of **q**-**polynomials** from the Askey scheme, which contain the independent variable x in one of the parameters of the corresponding basic hypergeometric series.The simplest example of this type is the discrete **q**-Hermite II **polynomials**

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us to draw an analogy between the basic properties of the Clebsch-Gordan coecients and these orthogonal **q**- **polynomials**. Since these coecients are studied from view point of the theory of orthogonal **polynomials**, a group-theoretical interpretation arises for the basic properties of dual Hahn **q**-**polynomials**. In section 5 we nd the relation between Clebsch- Gordan coecients for the quantum algebra SU **q** (1 ; 1) and the dual Hahn **q**-**polynomials** by

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Another point of viewwas developed by the Russian (former Soviet) school of mathematicians starting from a work by Nikiforov and Uvarov in 1983 [27]. It was based on the idea that the **q**-**polynomials** are the solution of a second-order linear di<erence equation with certain properties: the so-called di<erence equation of hypergeometric type on non-uniform lattices. This scheme is usually called the Nikiforov–Uvarov scheme of **q**-**polynomials** [28]. For several surveys on this approach see [3,4,7,26,29].

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Suslov, The theory of dierene analogues of speial funtions of hypergeometri. type[r]

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of s. Here f (s) = f (s + 1) − f (s) and ∇ f (s) = f (s) − f (s − 1) denote the forward and backward difference operators, respectively. One of the properties of the above equation is that its polynomial solutions can be expressed as basic hypergeometric series. In particular, when the lattice function is x(s) = **q** s it becomes into the Hahn **q**-difference equation (1.9). This approach based on the difference equation is usually called the Nikiforov–Uvarov scheme of **q**-**polynomials** [26] (for more details see e.g., [3,6,24,27]).

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(a small subset of the **q**-world). Here we need to point out that exits two dierent point of view in the study of the **q**-**polynomials**. The rst one, in the framework of the **q**-basic hypergeometric series [6], [8], [9] and the second, in the framework of the theory of dierence equations developed by Nikiforov et al. [12], [13], [14]. In this work we will use the second one because it gives us the possibility to provide an uniform treatment of several classes of orthogonal **polynomials** and, probably, it is the best way to nd further applications.

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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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In this paper we study in detail a **q**-extension of the generalized Hermite **polynomials** of Szeg˝ o. A continuous orthogonality property on R with respect to the positive weight function is established, a **q**-difference equation and a three-term recurrence relation are derived for this family of **q**-**polynomials**.

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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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tion of the non-standard **q**-Racah **polynomials** was considered in detail. This is an important example for two reasons: 1) it is the first family of the Krall- type **polynomials** on a non-linear type lattice that has been studied in detail and 2) almost all modifications (via the addition of delta Dirac masses) of the classical and **q**-classical **polynomials** can be obtained from them by taking appropriate limits (as it is shown for the dual **q**-Hahn, the Racah, and the **q**- Hahn **polynomials** in section 4.2). Let us also mention here that an instance of the Krall-type **polynomials** obtained from the Askey-Wilson **polynomials** (with a certain choice of parameters), by adding two mass points at the end of the orthogonality has been mentioned in [20, §6, page 330]. This Askey-Wilson-Krall-type **polynomials** solve the so-called bi-spectral problem associated with the Askey-Wilson operator. Then, it is an interesting open problem to study the general Krall-type Askey-Wilson **polynomials** and to obtain their main properties. This will be considered in a forthcoming paper. Acknowledgements:

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In [P] Pisier gave a counterexample to a well-known conjecture of Grothendieck [G] regarding the non-existence of non-nuclear locally convex spaces E and F such that the injective and projective tensor norms coincide on the tensor product E F . Pisier constructed a Banach space P such that P ε P P π P. Thus, over Pisier’s space all 2-homogeneous **polynomials** are integral. The situation is very diﬀerent for **polynomials** of degree higher than 2. Indeed, [Pg] has shown that any inﬁnite-dimensional Banach space admits extendible non-integral **polynomials** of any degree higher that 3, and in [CD] Carando and Dimant close the gap by proving the same for any degree higher than 2. Their construction makes use of ﬁnite-dimensional estimates of Boas [Boa].

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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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parameters (all Bernstein **polynomials** are zero but extreme **polynomials**). This way, global error is almost neglected and numerical solution is practi- cally exact. This property is not accomplished by MLS-based EFGM because truncation of shape functions is obtained at boundaries. Besides, some parameters could complicate the selection of the best MLS approximation, such as weight functions, support radius, or the disappearance of order of the polynomial base of the analysis. Finally, from the computing point of view, Bernstein curves are poly- nomials, which are a PU; so computing them gives low-cost shape functions.

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Key words and phrases: disrete Chebyshev polynomials, speeh reognition, speakerr. veriation, speeh identiation, speeh sound, speeh ompression.[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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index. In addition, we will work with families of classics multiple orthogonal **polynomials** according to the classification in [1] given by weight functions which are solution of the Pearson’s equation in Angelesco Systems and AT- systems. We also assume the results in a weakly perfect system where multiple orthogonal type II **polynomials** are monic.

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Mixtures of polynomials (MoPs), 1,2 mixtures of truncated basis functions (MoTBFs), 3 and mixtures of truncated exponentials (MTEs) 4 have recently been proposed as nonparametric de[r]

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ing orthogonal **polynomials** with respect to the inner product (2), usually called the discrete Sobolev-type Laguerre which constitutes an instance of a larger class of orthogonal polynomi- als: the discrete Sobolev-type orthogonal **polynomials**. For more detailed description of this Sobolev-type orthogonal **polynomials** (including the continuous ones) we refer the readers to the recent reviews [12, 14, 15].

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