PDF superior A characterization of the classical orthogonal discrete and q-polynomials

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.

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

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.)

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.

of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures supported in [ 1 ; 1] and satisfying some extra conditions in terms of the parameters of the three term recurrence relation which the corresponding sequence of orthogonal polynomials satisfy.

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].

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.

The structure of the paper is the following. In Section 2 we give some results concerning to classical Jacobi polynomials. Using these results, in Section 3 we obtain an explicit formula for the Jacobi- Sobolev-type orthogonal polynomials in terms of the classical ones and their rst and second derivatives which allows us to deduce a symmetry property. In Section 4 we establish the recurrence relation that the Jacobi-Sobolev-type orthogonal polynomials satisfy, when the masses A

The term moment problem was used for the first time in T. J. Stieltjes’ clas- sic memoir [32] (published posthumously between 1894 and 1895) dedicated to the study of continued fractions. The moment problem is a question in classical analysis that has produced a rich theory in applied and pure mathematics. This problem is beautifully connected to the theory of orthogonal polynomials, spectral representation of operators, matrix factorization problems, probability, statistics, prediction of stochastic processes, polynomial optimization, inverse problems in financial mathematics and function theory, among many other areas. In the ma- trix case, M. Krein was the first to consider this problem in [21], and later on some density questions related to the matrix moment problem were addressed in [14, 15, 24, 25]. Recently, the theory of the matrix moment problem is used in [10] for the analysis of random matrix-valued measures. Since the matrix moment problem is closely related to the theory of matrix orthogonal polynomials, M. Krein was the first to consider these polynomials in [22]. Later, several researchers have made contributions to this theory until today. In the last 30 years, several known properties of orthogonal polynomials in the scalar case have been extended to the matrix case, such as algebraic aspects related to their zeros, recurrence relations, Favard type theorems, and Christoffel–Darboux formulas, among many others.

The modication of classical functionals have been considered also for the discrete orthogo- nal polynomials. In this direction Bavinck and van Haeringen [9] obtained an innite order dierence equation for generalized Meixner polynomials, i.e., polynomials orthogonal with respect to the modication of the Meixner weight with a point mass at x = 0. The same was found for generalized Charlier polynomials by Bavinck and Koekoek [10]. In a series of papers by Alvarez-Nodarse et. al [2]-[4] the authors have obtained the representation as hy- pergeometric functions for generalized Meixner, Charlier, Kravchuk and Hahn polynomials as well as the corresponding second order dierence equation that such polynomials satisfy. The connection of all these discrete polynomials with the Jacobi [27] and Laguerre [21] type where studied in details in [5]. In particular, in [5] they proved that the Jacobi-Koornwinder polynomials [27] are a limit case of the generalized Hahn as well as the Laguerre-Koekoek [21], [23] are of the Meixner ones.

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).

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.

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]

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]).

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]

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

The structure of the paper is as follows. In Section 2 we list some of the main properties of the classical Bessel polynomials which will be used later on. In Section 3 we dene the generalized polynomials and nd some of their properties. In Section 4 we obtain the representation of the generalized Bessel polynomials in terms of the hypergeometric functions. In Section 5 we obtain an asymptotic formula for these polynomials and in Section 6 we establish their quasi-orthogonality. Finally, in Sections 7 and 8 we obtain the three-term recurrence relation that such polynomials satisfy as well as the corresponding Jacobi matrices.

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.

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.

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