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R. Alvarez-Nodarse

Publications and source records attributed to R. Alvarez-Nodarse.

10 recordsLinked to original sources
arXiv2025

On the quantum algebra $su_q(1,1)$ from a Special Function standpoint

In this paper, we study the tensor product of two unitary irreducible representations, as well as the tensor product of a unitary irreducible representation with a finite-dimensional one, and determine the corresponding Clebsch-Gordan coefficients. Using von Neumann's projection operator method, we obtain an explicit representation of these coefficients, which allows us to express them as symmetric q-hypergeometric series. Finally, by leveraging the properties of the q-hypergeometric function, we derive several properties of the Clebsch-Gordan coefficients, including a number of new results, in a unified and straightforward manner.

math-ph↗
arXiv2025

Oscillatory instability and stability of stationary solutions in the parametrically driven, damped nonlinear Schrödinger equation

We found two stationary solutions of the parametrically driven, damped nonlinear Schrödinger equation with nonlinear term proportional to $|ψ(x,t)|^{2 κ} ψ(x,t)$ for positive values of $κ$. By linearizing the equation around these exact solutions, we derive the corresponding Sturm-Liouville problem. Our analysis reveals that one of the stationary solutions is unstable, while the stability of the other solution depends on the amplitude of the parametric force, damping coefficient, and nonlinearity parameter $κ$. An exceptional change of variables facilitates the computation of the stability diagram through numerical solutions of the eigenvalue problem as a specific parameter $\varepsilon$ varies within a bounded interval. For $κ<2$ , an {\it oscillatory instability} is predicted analytically and confirmed numerically. Our principal result establishes that for $κ\ge 2$, there exists a critical value of $\varepsilon$ beyond which the unstable soliton becomes stable, exhibiting {\it oscillatory stability}.

nlin.PS↗
arXiv2014

On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Let D_v the difference operator and q-difference operators defined by D_ωp(x) = \frac{p(x+ω)-p(x)}ω and D_q p(x) = \frac{p(qx)-p(x)}{(q-1)x}, respectively. Let U and V be two moment regular linear functionals and let (P_n)_n and Q_n)_n be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the above two OPS assuming that their difference derivatives $D_ν$ of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as $$ \sum_{i=0}^M a_{i,n} D_ν^m P_{n+m-i}(x) = \sum_{i=0}^N b_{i,n} D_ν^k Q_{n+k-i}(x), \quad n\geq 0, $$ where $M,N,m,k=0,1,2,... Under certain conditions, we prove that U and V are related by a rational factor ç (in the distributional sense). Moreover, when m\neq k then both U and V are D_v-semiclassical functionals. This leads us to the concept of (M,N)-D_v-coherent pair of order (m,k) extending to the discrete case several previous works. As an application we consider the OPS with respect to a certain following Sobolev-type discrete inner product.

math.CA↗
arXiv2012

On the limit of non-standard q-Racah polynomials

The aim of this article is to study the limit transitions from non-standard q-Racah polynomials to big q-Jacobi, dual q-Hahn, and q-Hahn polynomials such that the orthogonality properties and the three-term recurrence relations remain valid.

math.CA↗
arXiv2012

The orthogonality of q-classical polynomials of the Hahn class: A geometrical approach

The idea of this review article is to discuss in a unified way the orthogonality of all positive definite polynomial solutions of the $q$-hypergeometric difference equation on the $q$-linear lattice by means of a qualitative analysis of the $q$-Pearson equation. Therefore, our method differs from the standard ones which are based on the Favard theorem, the three-term recurrence relation and the difference equation of hypergeometric type. Our approach enables us to extend the orthogonality relations for some well-known $q$-polynomials of the Hahn class to a larger set of their parameters. A short version of this paper appeared in SIGMA 8 (2012), 042, 30 pages http://dx.doi.org/10.3842/SIGMA.2012.042.

math.CA↗
arXiv2012

The q-Racah-Krall-type polynomials

In this paper the Krall-type polynomials obtained via the addition of two mass points to the weight function of the \textit{standard} $q$-Racah polynomials are introduced. Several algebraic properties of these polynomials are obtained and some of their limit cases are discussed.

math.CA↗
arXiv2012

On the Krall-type Askey-Wilson Polynomials

In this paper the general Krall-type Askey-Wilson polynomials are introduced. These polynomials are obtained from the Askey-Wilson polynomials via the addition of two mass points to the weight function of them at the points $\pm1$. Several properties of such new family are considered, in particular the three-term recurrence relation and the representation as basic hypergeometric series.

math.CA↗
arXiv2012

On the orthogonality of q-classical polynomials of the Hahn class II

In this article, the study of the orthogonality properties of $q$-polynomials of the Hahn class started in the initial article by R. Álvarez-Nodarse, R. Sevinik-Adıgüzel, and H. Taşeli, \textit{On the orthogonality of $q$-classical polynomials of the Hahn class I} is proceeded. To be more specific, the orthogonality properties of the $q$-polynomials belonging to the $\emptyset$-Hermite-Laguerre/Jacobi, $\emptyset$-Jacobi/Hermite-Laguerre, 0-Laguerre/Jacobi-Bessel and 0-Jacobi/Laguerre-Bessel cases are studied by taking into account the idea considered in the initial paper. In particular, a new orthogonality relation for the $q$-Meixner polynomials is established.

math.CA↗
arXiv2011

On the Properties of Special Functions on the linear-type lattices

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to $q$-classical polynomials are given.

math.CA↗