Thomas Dreyfus
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View article: Degenerate systems of three Brownian particles with asymmetric collisions: invariant measure of gaps
Degenerate systems of three Brownian particles with asymmetric collisions: invariant measure of gaps Open
We consider a degenerate system of three Brownian particles undergoing asymmetric collisions. We study the gap process of this system and focus on its invariant measure. The gap process is described as an obliquely reflected degenerate Bro…
View article: On the product of two generic $q$-Gevrey series
On the product of two generic $q$-Gevrey series Open
In this paper, we consider a $q$-analog of the Borel-Laplace summation process introduced by Fabienne Marotte and the second author. We specifically examine two power series solutions of linear $q$-difference equations whose Newton polygon…
View article: Enumeration of weighted quadrant walks: criteria for algebraicity and D-finiteness
Enumeration of weighted quadrant walks: criteria for algebraicity and D-finiteness Open
In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on t…
View article: Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation
Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation Open
In the current paper we study the -analogue introduced by Jimbo and Sakai of the well known Painlevé VI differential equation. We explain how it can be deduced from a -analogue of Schlesinger equations and show that for a convenient change…
View article: On the product of two $1$-$q$-summable series
On the product of two $1$-$q$-summable series Open
In this paper we consider a $q$-analog of the Borel-Laplace summation process defined by Marotte and the second author, and consider two series solutions of linear $q$-difference equations with slopes $0$ and $1$. The latter are $q$-summab…
View article: Algebraic independence and linear difference equations
Algebraic independence and linear difference equations Open
We consider pairs of automorphisms (\phi,\sigma) acting on fields of Laurent or Puiseux series: pairs of shift operators (\phi\colon x\mapsto x+h_1, \sigma\colon x\mapsto x+h_2) , of q -difference operators (\phi\colon x\mapsto q_1x , \sig…
View article: Hypertranscendence and linear difference equations, the exponential case
Hypertranscendence and linear difference equations, the exponential case Open
In this paper we study meromorphic functions solutions of linear shift difference equations in coefficients in $\mathbb{C}(x)$ involving the operator $ρ: y(x)\mapsto y(x+h)$, for some $h\in \mathbb{C}^*$. We prove that if $f$ is solution o…
View article: On the computation of the difference Galois groups of order three equations
On the computation of the difference Galois groups of order three equations Open
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 oper…
View article: Length derivative of the generating function of walks confined in the quarter plane
Length derivative of the generating function of walks confined in the quarter plane Open
In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series that count the number of walks conf…
View article: Differential algebraic generating series of weighted walks in the quarter plane
Differential algebraic generating series of weighted walks in the quarter plane Open
In the present paper we study the nature of the trivariate generating series of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies…
View article: Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions
Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions Open
We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, -di…
View article: Hypertranscendence and linear difference equations
Hypertranscendence and linear difference equations Open
After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental (i.e., they cannot be solution to an algebraic…
View article: Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation
Degeneration from difference to differential Okamoto spaces for the sixth Painlevé equation Open
In the current paper we study the $q$-analogue introduced by Jimbo and Sakai of the well known Painlevé VI differential equation. We explain how it can be deduced from a $q$-analogue of Schlesinger equations and show that for a convenient …
View article: Whose Consent is it Anyway? A Transgender Child's Right to Transition
Whose Consent is it Anyway? A Transgender Child's Right to Transition Open
A landmark decision delivered by the Full Court of the Family Court of Australia on November 30th, 2017 confirmed that a transgender child who is capable of giving informed consent can undertake gender affirming hormone treatment without a…
View article: On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients
On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients Open
The analytic and formal solutions of certain family of\n$q$-difference-differential equations under the action of a complex\nperturbation parameter is considered. The previous study of the last two\nauthors provides information in the case…
View article: Computing the Lie algebra of the differential Galois group: the reducible case
Computing the Lie algebra of the differential Galois group: the reducible case Open
In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-K…
View article: Length derivative of the generating series of walks confined in the quarter plane
Length derivative of the generating series of walks confined in the quarter plane Open
In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of …
View article: Computing solutions of linear Mahler equations
Computing solutions of linear Mahler equations Open
Mahler equations relate evaluations of the same function $f$ at iterated $b$th powers of the variable. They arise in particular in the study of automatic sequences and in the complexity analysis of divide-and-conquer algorithms. Recently, …
View article: Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks
Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks Open
We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain…
View article: Twofold $q$-Gevrey asymptotics for linear singularly perturbed $q$-difference-differential equations with polynomial coefficients
Twofold $q$-Gevrey asymptotics for linear singularly perturbed $q$-difference-differential equations with polynomial coefficients Open
We construct analytic and formal solutions for a family of $q$-difference-differential problems, under the action of a perturbation parameter. This work is a continuation of a previous work focusing on a singularly perturbed $q$-difference…
View article: <i>q</i>-Borel-Laplace summation for <i>q</i>–difference equations with two slopes
<i>q</i>-Borel-Laplace summation for <i>q</i>–difference equations with two slopes Open
Ce papier a été initialement écrit sous le titre suivant : "Computing Bugeaud's solutions of q-difference equations with a q-Borel-Laplace summation"