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Displaying 41 – 55 of 55

q-Heat Operator and q-Poisson’s Operator

Mabrouk, Hanène (2006)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 33D15, 33D90, 39A13In this paper we study the q-heat and q-Poisson’s operators associated with the q-operator ∆q (see[5]). We begin by summarizing some statements concerning the q-even translation operator Tx,q, defined by Fitouhi and Bouzeffour in [5]. Then, we establish some basic properties of the q-heat semi-group such as boundedness and positivity. In the second part, we introduce the q-Poisson operator P^t, and address its main properties. We show...

Representations of quantum groups and (conditionally) invariant q-difference equations

Vladimir Dobrev (1997)

Banach Center Publications

We give a systematic discussion of the relation between q-difference equations which are conditionally $U_{q} ()$ -invariant and subsingular vectors of Verma modules over $U_{q} ()$ (the Drinfeld-Jimbo q-deformation of a semisimple Lie algebra over ℂg or ℝ). We treat in detail the cases of the conformal algebra, = su(2,2), and its complexification, = sl(4). The conditionally invariant equations are the q-deformed d’Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel’fand-Gel’fand....

Second order linear $q$ -difference equations: nonoscillation and asymptotics

Pavel Řehák (2011)

Czechoslovak Mathematical Journal

The paper can be understood as a completion of the $q$ -Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$ -difference equations. The $q$ -Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $q^{ℕ_{0}} : = {q^{k} : k \in ℕ_{0}}$ with $q > 1$ . In addition to recalling the existing concepts of $q$ -regular variation and $q$ -rapid variation we introduce $q$ -regularly bounded functions and prove many related properties. The $q$ -Karamata theory is then...

Séries de $q$ -factorielles, opérateurs aux $q$ -différences et confluence

Anne Duval (2003)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Singular boundary value problem on infinite time scale.

Hao, Zhao-Cai, Liang, Jin, Xiao, Ti-Jun (2006)

Discrete Dynamics in Nature and Society

Some properties of solutions of complex q-shift difference equations

Hong-Yan Xu, Jin Tu, Xiu-Min Zheng (2013)

Annales Polonici Mathematici

Combining difference and q-difference equations, we study the properties of meromorphic solutions of q-shift difference equations from the point of view of value distribution. We obtain lower bounds for the Nevanlinna lower order for meromorphic solutions of such equations. Our results improve and extend previous theorems by Zheng and Chen and by Liu and Qi. Some examples are also given to illustrate our results.

Spectral theory from the second-order $q$ -difference operator.

Dhaouadi, Lazhar (2007)

International Journal of Mathematics and Mathematical Sciences

Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

Bilender P. Allahverdiev, Hüseyin Tuna (2020)

Communications in Mathematics

In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

Stability of nonlinear $h$ -difference systems with $n$ fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

In the paper we study the subject of stability of systems with $h$ -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$ -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$ -orders.

Symmetries in connection preserving deformations.

Ormerod, Christopher M. (2011)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Systèmes aux $q$ -différences singuliers réguliers : classification, matrice de connexion et monodromie

Jacques Sauloy (2000)

Annales de l'institut Fourier

G.D. Birkhoff a posé, par analogie avec le cas classique des équations différentielles, le problème de Riemann-Hilbert pour les systèmes “fuchsiens” aux $q$ -différences linéaires, à coefficients rationnels. Il l’a résolu dans le cas générique: l’objet classifiant qu’il introduit est constitué de la matrice de connexion $P$ et des exposants en $0$ et $\infty$ . Nous reprenons sa méthode dans le cas général, mais en traitant symétriquement $0$ et $\infty$ et sans recours à des solutions à croissance “sauvage”. Lorsque $q$ ...

The convolution on time scales.

Bohner, Martin, Guseinov, Gusein Sh. (2007)

Abstract and Applied Analysis

The lattice structure of connection preserving deformations for $q$ -Painlevé equations I.

Ormerod, Christopher M. (2011)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

The rational qKZ equation and shifted non-symmetric Jack polynomials.

Kakei, Saburo, Nishizawa, Michitomo, Saito, Yoshihisa, Takeyama, Yoshihiro (2009)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations

Luc Haine (2005)

Annales de l’institut Fourier

The Stieltjes spectral matrix measure of the doubly infinite Jacobi matrix associated with a Toda $g$ -soliton is computed, using Sato theory. The result is used to give an explicit expansion of the fundamental solution of some discrete heat equations, in a series of Jackson’s $q$ -Bessel functions. For Askey-Wilson type solitons, this expansion reduces to a finite sum.

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