Related papers: Functional relations of solutions of q-difference …
In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely…
The second order hypergeometric q-difference operator is studied for the value c=-q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space l^2(Z). The operator has…
The local analytic classification and the description of the Galois group for complex linear analytic q-difference equations have been obtained by Ramis, Sauloy and Zhang [15, 14] under the assumption that the slopes of the Newton polygon…
Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all…
This thesis is concerning to the Differential Galois Theory point of view of the Supersymmetric Quantum Mechanics. The main object considered here is the non-relativistic stationary Schr\"odinger equation, specially the integrable cases in…
We consider a family of solutions of $q-$difference Riccati equation, and prove the meromorphic solutions of $q-$difference Riccati equation and corresponding second order $q-$difference equation are concerning with $q-$gamma function. The…
We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over arbitrary algebraic number fields. Our approach combines constructions of type II Pad\'{e}…
In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the $q$-difference operator (Case Q), the Mahler…
In the present paper, a general theory for the second-order matrix difference equation of bilateral type is discussed. We introduced the matrix $q$-Kummer equation of bilateral type and presented the $q$-Kummer matrix function as a series…
We study the basic Galois connection induced by the "satisfaction" relation between external operations $A^n\rightarrow B$ defined on a set $A$ and valued in a possibly different set $B$ on the one hand, and ordered pairs $(R,S)$ of…
Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved…
We prove orthogonality relations for some analogs of trigonometric functions on a $q$-quadratic grid and introduce the corresponding $q$-Fourier series. We also discuss several other properties of this basic trigonometric system and the…
We investigate the integral representations of solutions to the variant of $q$-hypergeometric equation of degree 2 obtained through $q$-middle convolution by using transformation formulas for $q$-hypergeometric series. We show the…
Choose $q\in {\mathbb C}$ with 0<|q|<1. The main theme of this paper is the study of linear q-difference equations over the field K of germs of meromorphic functions at 0. It turns out that a difference module M over K induces in a…
We present a Galois theory connecting finitary operations with pairs of finitary relations one of which is contained in the other. The Galois closed sets on both sides are characterised as locally closed subuniverses of the full iterative…
In this paper, we establish the linear independence of values of the $q$-analogue of the exponential function, $E_q(x)$ and its derivatives at specified algebraic arguments, when $q$ is a Pisot-Vijayraghavan number. We also deduce similar…
We apply Rossi's half-plane version of Borel's Theorem to study the zero distribution of linear combinations of $\mathcal{A}$-entire functions (Theorem 1.2). This provides a unified way to study linear $q$-difference, difference and…
We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2\psi_2$ series, they are…
We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of Q-linearly independent algebraic numbers are algebraically independent) for commutative algebraic groups G without unipotent quotients, over function…
We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between…