A Darboux-type theorem for slowly varying functions

AbstractFor functionsg(z) satisfying a slowly varying condition in the complex plane, we find asymptotics for the Taylor coefficients of the function[formula]whenα>0. As applications we find asymptotics for the number of permutations with cycle lengths all lying in a given setS, and for the number having unique cycle lengths

A solution to the Al-Salam--Chihara moment problem

We study the $q$-hypergeometric difference operator $L$ on a particular Hilbert space. In this setting $L$ can be considered as an extension of the Jacobi operator for $q^{-1}$-Al-Salam--Chihara polynomials. Spectral analysis leads to unitarity and an explicit inverse of a $q$-analog of the Jacobi function transform. As a consequence a solution of the Al-Salam--Chihara indeterminate moment problem is obtained.Comment: 22 page

Symmetric diffusions with polynomial eigenvectors

25 pagesInternational audienceWe describe symmetric diffusion operators where the spectral decomposition is given through a family of orthogonal polynomials. In dimension one, this reduces to the case of Hermite, Laguerre and Jacobi polynomials. In higher dimension, some basic examples arise from compact Lie groups. We give a complete description of the bounded sets on which such operators may live. We then provide a classification of those sets when the polynomials are ordered according to their usual degree

Transseries for a class of nonlinear difference equations

Given a nonlinear analytic difference equation of level I with a formal power series solution (y) over cap (0) we associate with it a stable manifold of solutions with asymptotic expansion (y) over cap (0). This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion (y) over cap (0) in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]

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 x n-matrix meromorphic in a neighborhood of infinity with det A(s) not equal 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 analytic fundamental matrices represented asymptotically by the formal fundamental solution in appropriate domains. The method of proof is analogous to a method used to prove multi-summability of formal solutions of ODE's. Starting from analytic lifts of the formal fundamental matrix in half planes, we construct a sequence of increasingly precise quasi-functions, each of which is determined uniquely by its predecessor