Sur les équations fonctionnelles aux itérées

Abstract

Summary. In this paper, we study the convergence of formal power series solutions ~ of functional equations of the form ~ Pk(X)~b(q~lk~(x)) = O(X), where t, olkl(x) denotes the k-th iterate of the function ~.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differemial equations: if ~p(0) = 0, and q~'(0) = q is a nonzero complex number with absolute value less than one then, if ~p(x) = ~ a(n)x" is a divergent solution, there exists a positive real number s such that the power series ~ a(n)q '"~"+ ~) 2x" has a finite and nonzero radius of convergence.

On q-summation and confluence

“…, and N q is fuchsian at 0 and ∞ (4) . (3) The Newton-Ramis polygons of N q coincide for any q ∈ (η, 1], and the coefficients A i (q, ξ) tends uniformly to A i (1, ξ) when q → 1, on any compact of P 1 C .…”

Confluence of meromorphic solutions of q-difference equations

“…Due to (A2), the slopes of the σ q -equation satisfied byĥ are independent of q, and the smallest positive slope is k 1 . As we can see in [32], Theorem 4.8, (see also [6]), there exist C 1 (q), C 2 (q) > 0, such that for all l ∈ N, for all q > 1…”

Recent results on functional equations in a single variable, perspectives and open problems

“…The following result due to J.-P. Bézivin [41] serves as an example. Concerning a similar problem for other equations and fields see [44] by J.-P. Bé-zivin and A. Boutabaa and [42] and [43] by J.-P. Bézivin.…”

“…with a non-zero polynomial P , considered by J.-P. Bézivin [40] who proved, among others, that every entire solution of (3.5) has to be a polynomial. Many people were interested in the functional equation …”