1. CJM 2016 (vol 69 pp. 687)
 Ovchinnikov, Alexey; Wibmer, Michael

Tannakian Categories with Semigroup Actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are
algebraically independent over $\mathbb{C}(x)$. More generally, for
a linear differential or difference equation, it is an important
problem to find all algebraic dependencies among a nonzero solution
$y$ and particular transformations of $y$, such as derivatives
of $y$ with respect to parameters, shifts of the arguments, rescaling,
etc. In the present paper, we develop a theory of Tannakian categories
with semigroup actions, which will be used to attack such questions
in full generality, as each linear differential equation gives
rise to a Tannakian category.
Deligne studied actions of braid groups on categories and obtained
a finite collection of axioms that characterizes such actions
to apply it to various geometric constructions. In this paper,
we find a finite set of axioms that characterizes actions of
semigroups that are finite free products of semigroups of the
form $\mathbb{N}^n\times
\mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$
on Tannakian categories. This is the class of semigroups that
appear in many applications.
Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06 

2. CJM 2015 (vol 67 pp. 1270)
 Carcamo, Cristian; Vidal, Claudio

Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems
In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.
Keywords:difference equations, Hamiltonian systems, stability in the Lyapunov sense Categories:34D20, 34E10 

3. CJM 2014 (vol 67 pp. 1065)
4. CJM 2002 (vol 54 pp. 709)
 Ismail, Mourad E. H.; Stanton, Dennis

$q$Integral and Moment Representations for $q$Orthogonal Polynomials
We develop a method for deriving integral representations of certain
orthogonal polynomials as moments. These moment representations are
applied to find linear and multilinear generating functions for
$q$orthogonal polynomials. As a byproduct we establish new
transformation formulas for combinations of basic hypergeometric
functions, including a new representation of the $q$exponential
function $\mathcal{E}_q$.
Keywords:$q$integral, $q$orthogonal polynomials, associated polynomials, $q$difference equations, generating functions, AlSalamChihara polynomials, continuous $q$ultraspherical polynomials Categories:33D45, 33D20, 33C45, 30E05 
