On discrete q-ultraspherical polynomials and their duals

We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q), which corresponds to a=b=-c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a< q^{-1}). Since P_n(x;q^\alpha, q^\alpha, -q^\alpha; q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q->1, this family represents yet another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. The dual family with respect to the polynomials P_n(x;a,a,-a;q) (i.e., the dual discrete q-ultraspherical polynomials) corresponds to the indeterminate moment problem, that is, these polynomials have infinitely many orthogonality relations. We find orthogonality relations for these polynomials, which have not been considered before. In particular, extremal orthogonality measures for these polynomials are derived.Comment: 14 page

Jacobi Matrix Pair and Dual Alternative q-Charlier Polynomials

By using two operators representable by Jacobi matrices, we introduce a family of q-orthogonal polynomials, which turn out to be dual with respect to alternative q-Charlier polynomials. A discrete orthogonality relation and the completeness property for these polynomials are established.3a допомогою двох операторів, зображуваних матрицями Якобі, введено сім'ю q-ортогональних многочленів, що є дуальними по відношенню до альтернативних q-многочленів Шарльє. Для цих многочленів отримано дискретне співвідношення ортогональності та властивість повноти

qq-Classical orthogonal polynomials: A general difference calculus approach

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues operator are deduced. A more general characterization Theorem that the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn Tableaux, respectively, is established. Finally, the families of Askey-Wilson polynomials, q-Racah polynomials, Al-Salam & Carlitz I and II, and q-Meixner are considered. [1] R. Alvarez-Nodarse. On characterization of classical polynomials. J. Comput. Appl. Math., 196:320{337, 2006. [2] M. Alfaro and R. Alvarez-Nodarse. A characterization of the classical orthogonal discrete and q-polynomials. J. Comput. Appl. Math., 2006. In press.Comment: 18 page

Spectral properties of a generalized chGUE

We consider a generalized chiral Gaussian Unitary Ensemble (chGUE) based on a weak confining potential. We study the spectral correlations close to the origin in the thermodynamic limit. We show that for eigenvalues separated up to the mean level spacing the spectral correlations coincide with those of chGUE. Beyond this point, the spectrum is described by an oscillating number variance centered around a constant value. We argue that the origin of such a rigid spectrum is due to the breakdown of the translational invariance of the spectral kernel in the bulk of the spectrum. Finally, we compare our results with the ones obtained from a critical chGUE recently reported in the literature. We conclude that our generalized chGUE does not belong to the same class of universality as the above mentioned model.Comment: 12 pages, 3 figure

The Wigner function for general Lie groups and the wavelet transform

We build Wigner maps, functions and operators on general phase spaces arising from a class of Lie groups, including non-unimodular groups (such as the affine group). The phase spaces are coadjoint orbits in the dual of the Lie algebra of these groups and they come equipped with natural symplectic structures and Liouville-type invariant measures. When the group admits square-integrable representations, we present a very general construction of a Wigner function which enjoys all the desirable properties, including full covariance and reconstruction formulae. We study in detail the case of the affine group on the line. In particular, we put into focus the close connection between the well-known wavelet transform and the Wigner function on such groups

The Wigner function for general Lie groups and the wavelet transform

We build Wigner maps, functions and operators on general phase spaces arising from a class of Lie groups, including non-unimodular groups (such as the affine group). The phase spaces are coadjoint orbits in the dual of the Lie algebra of these groups and they come equipped with natural symplectic structures and Liouville-type invariant measures. When the group admits square-integrable representations, we present a very general construction of a Wigner function which enjoys all the desirable properties, including full covariance and reconstruction formulae. We study in detail the case of the affine group on the line. In particular, we put into focus the close connection between the well-known wavelet transform and the Wigner function on such groups