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On $$q$$- difference functional equations. (Sur les équations fonctionnelles aux $$q$$-différences.) (French) Zbl 0757.39002
As usual, let $$\mathbb{C}[[x]]$$ denote the field of formal power series in $$x$$ with coefficients in $$\mathbb{C}$$. Let $$G(\mathbb{C})$$ denote the space of germs of functions analytic at the origin and $$A(\mathbb{C})$$ the space of functions analytic on $$\mathbb{C}$$. Denote the Hadamard product by $$\square$$, and let $$\eta_ s(x)=\sum q^{s[n(n-1)/2]}x^ n$$. Denote by $$\mathbb{C}[[x]]_{q,s}$$ the space of series $$f=\sum a_ nx^ n$$ such that $$\eta_ s\square f\in G(\mathbb{C})$$, and by $$C[[x]]_{q,(s)}$$, the analogous space in which $$G(\mathbb{C})$$ is replaced by $$A(\mathbb{C})$$. Denote by $$\varphi^{[i]}(x)$$ the $$i$$th iterate of the function $$\varphi(x)$$.
Consider the functional equation $$\sum^ s_{i=0}P_ i(x)\psi(\varphi^{[i]}(x))=\theta(x)$$, where the $$P_ i(x)$$, $$\varphi(x)$$, and $$\theta(x)$$ have non-zero radius of convergence. The author’s principal result is that under these conditions, if a formal solution $$\psi$$ exists then either $$\psi$$ has a nonzero radius of convergence, or $$s$$ is such that $$\psi$$ belongs to $$\mathbb{C}[[x]]_{q,s}$$ but not to $$C[[x]]_{q,(s)}$$. This latter value of $$s$$ is unique if it exists.
The proof proceeds by reducing the result to the case where $$\varphi(x)=qx$$, or, more precisely, the slightly more general situation of equations $$\sum P_ i(x)\psi(xq^{k_ i})=\theta(x)$$ where $$\{k_ i\}$$ is a sequence of reals tending to infinity.

##### MSC:
 39B12 Iteration theory, iterative and composite equations 39B32 Functional equations for complex functions 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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##### References:
  Adams, C. R.,On the linear ordinary q-difference equations. Ann. of Math. (2)30 (1929), 195–205. · JFM 55.0263.01  Birkhoff, G. D.,The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations. Proc. Amer. Acad.49 (1913), 521–568. · JFM 44.0391.03  Birkhoff, G. D. etGuenther, P. E.,Note on a canonical form for the linear q-difference equation. Proc. Nat. Acad. Sci. U.S.A.27 (1941), 218–222. · Zbl 0061.20002  Carmichael, R. D.,The general theroy nof linear q-difference equations. Amer. J. Math.34 (1912), 146–168. · JFM 43.0411.02  Duval, A.,Lemmes de Hensel et factorisation formelle pour les opérateurs aux différences. Funkcial. Ekvac.26 (1983), 349–368. · Zbl 0543.12018  Grevy, A.,Thèse. Gauthier-Villars, Paris, 1894 (publiée en grande partie dans Ann. Sci. Ecole Norm. Sup.11 (1894), 249–323).  Kuczma, M.,Functional equations in a single variable. Polish Scientific Publ., Warszawa-Katowice, 1968. · Zbl 0196.16403  Kuczma, M., Choczewski, B. etGer, R.,Iterative functional equations. [Encyclopedia Math. Appl., Vol. 32]. Cambridge Univ. Press, Cambridge–New York, 1990. · Zbl 0703.39005  Leau, L.,Etude sur les équations fonctionnelles. (Thèse). Paris, 1897. · JFM 25.0603.01  Malgrange, B.,Sur les points singuliers des équations différentielles. Enseign. Math.20 (1974), 147–176. · Zbl 0299.34011  Praagman, C.,The formal classification of linear difference equations. Nederl-Akad. Wetensch. Indag. Math.45 (1983), 249–261. · Zbl 0519.39003  Praagman, C.,Stokes and Gevrey phenomena in relation to index theorems in the theory of meromorphic linear difference equations. Funkcial. Ekvac.29 (1986), 259–279. · Zbl 0618.39002  Ramis, J.-P.,Théorèmes d’indice Gevrey pour les équations différentielles ordinaires. [Mem. Amer. Math. Soc., No. 296]. A.M.S., Providence, RI, 1984.  Ramis, J.-P.,Dévissage Gevrey. Astérisque59–60 (1978), 173–204.  Siegel, C.,Iteration of analytic functions. Ann. of Math.43 (1942), 607–612. · Zbl 0061.14904  Smajdor, A etSmajdor, W.,On the existence and uniqueness of analytic solutions of a linear functional equations. Math. Z.98 (1967), 235–242. · Zbl 0166.12803  Trjitzinsky, W. J.,Analytic theory of linear q-difference equations. Acta Math.61 (1933), 1–38. · Zbl 0007.21103
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