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

On discrete q-ultraspherical polynomials and their duals

AbstractA special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q−1). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures

Пара матриць Якобі та дуальні альтернативні q-многочлени Шарльє

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.This work was performed during a visit of the second author (AUK) to the Institute of Mathematics, UNAM, Mexico. The participation of the first author (NMA) has been supported in part by the DGAPA-UNAM project IN102603-3 “Óptica Matemática”

Пара матриць Якобі та дуальні альтернативні q-многочлени Шарльє

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.This work was performed during a visit of the second author (AUK) to the Institute of Mathematics, UNAM, Mexico. The participation of the first author (NMA) has been supported in part by the DGAPA-UNAM project IN102603-3 “Óptica Matemática”

On q-orthogonal polynomials, dual to little and big q-Jacobi polynomials

AbstractWe derive discrete orthogonality relations for polynomials, dual to little and big q-Jacobi polynomials. This derivation essentially requires use of bases, consisting of eigenvectors of certain self-adjoint operators, which are representable by a Jacobi matrix. Recurrence relations for these polynomials are also given