Summation of Formal Solutions of a Class of Linear Diierence Equations

Q-hypergeometric Solutions of Q-diierence Equations

We present algorithm qHyper for nding all solutions y(x) of a linear homogeneous q-diierence equation such that y(qx) = r(x)y(x) where r(x) is a rational function of x. Applications include construction of basic hypergeometric series solutions, and deenite q-hypergeometric summation in closed form.

Numerical Solutions of Special Class of Systems of Non-Linear Volterra Integro-Differential Equations by a Simple High Accuracy Method

On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s

In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along ...

Multi-level Gevrey solutions of singularly perturbed linear partial differential equations

We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a BorelLaplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analyti...

Summation of Formal Solutions of a Class of Linear Difference Equations

We consider difference equations y(s+1) = A(s)y(s), where A(s) is an n × n-matrix meromorphic in a neighborhood of ∞ with detA(s) ≡ 0. In general, the formal fundamental solutions of this equation involve gamma-functions which give rise to the critical variable s log s and a level 1. We show that, under a mild condition, formal fundamental matrices of the equation can be summed uniquely to anal...