3 Analytic classification of admissible q-difference modules 10 3.1 Generalities on q-difference modules . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Simple objects in the category of admissible q-difference modules . . . . . . . 11 3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Analytic vs formal classification . . . . . . . . . . . . ...

In this work we give sufficient conditions in order that the difference equation with advanced arguments x(n + 1) = A(n)x(n) + B(n)x(σ1(n)) + f(n, x(n), x(σ2(n))), σi(n) ≥ n + 1 has Φ−bounded solutions in the space `Φ of Φ−bounded sequences, are given.

The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space around these fractal trees. We present the canonical and degenerate form of this fractal space and extend the fractal geometrical space to R explicitly and R ...

This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.

We introduce a group theoretical tool on which one can derive a family of identities from sequences that are defined by a recursive relation. As an illustration it is shown that n−1 ∑ i=1 Fn−iF 2 i = 1 2 n ∑ i=1 (−1)(F2i − Fi) = ( Fn+1 2 )

The primary aim of this paper is to (provide tools in order to) compute Galois groups of classical irregular q-difference equations. We are particularly interested in quantizations of omnipresent differential equations in the mathematical and physical literature, namely confluent generalized q-hypergeometric equations and q-Kloosterman equations.

We show that if v is a symmetric regular Laguerre-Hahn linear form (functional), then the linear form u defined by u = −λx−2v + δ0 is also regular and symmetric LaguerreHahn linear form for every complex λ except for a discrete set of numbers depending on v. We explicitly give the coefficients of the second-order recurrence relation, the structure relation of the orthogonal sequence associated ...

On the behavior of solutions of the system of rational difference equations x n+1 = x n−1 y n x n−1 − 1 , y n+1 = y n−1 x n y n−1 − 1 , z n+1 = 1 y n z n Abstract In this article, we investigate the solutions of the system of difference equations y n+1 = y n−1 x n y n−1 − 1 , y n+1 = y n−1 x n y n−1 − 1 , z n+1 = 1 y n z n where x 0 , x-1 , y 0 , y-1 , z 0 , z-1 real numbers such that y 0 x-1 ≠...

We present, for the first time, two hierarchies of nonlinear, integrable q-difference equations, one of which includes a q-difference form of each of the second and fifth Painlevé equations, qPII and qPV, the other includes qPIII. All the equations have multiple free parameters. A method to calculate a 2× 2 Lax pair for each equation in the hierarchy is also given.