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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.

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
Full Text: DOI EuDML
[1] Adams, C. R.,On the linear ordinary q-difference equations. Ann. of Math. (2)30 (1929), 195–205. · JFM 55.0263.01
[2] 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
[3] 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
[4] Carmichael, R. D.,The general theroy nof linear q-difference equations. Amer. J. Math.34 (1912), 146–168. · JFM 43.0411.02
[5] Duval, A.,Lemmes de Hensel et factorisation formelle pour les opérateurs aux différences. Funkcial. Ekvac.26 (1983), 349–368. · Zbl 0543.12018
[6] Grevy, A.,Thèse. Gauthier-Villars, Paris, 1894 (publiée en grande partie dans Ann. Sci. Ecole Norm. Sup.11 (1894), 249–323).
[7] Kuczma, M.,Functional equations in a single variable. Polish Scientific Publ., Warszawa-Katowice, 1968. · Zbl 0196.16403
[8] Kuczma, M., Choczewski, B. etGer, R.,Iterative functional equations. [Encyclopedia Math. Appl., Vol. 32]. Cambridge Univ. Press, Cambridge–New York, 1990. · Zbl 0703.39005
[9] Leau, L.,Etude sur les équations fonctionnelles. (Thèse). Paris, 1897. · JFM 25.0603.01
[10] Malgrange, B.,Sur les points singuliers des équations différentielles. Enseign. Math.20 (1974), 147–176. · Zbl 0299.34011
[11] Praagman, C.,The formal classification of linear difference equations. Nederl-Akad. Wetensch. Indag. Math.45 (1983), 249–261. · Zbl 0519.39003
[12] 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
[13] Ramis, J.-P.,Théorèmes d’indice Gevrey pour les équations différentielles ordinaires. [Mem. Amer. Math. Soc., No. 296]. A.M.S., Providence, RI, 1984.
[14] Ramis, J.-P.,Dévissage Gevrey. Astérisque59–60 (1978), 173–204.
[15] Siegel, C.,Iteration of analytic functions. Ann. of Math.43 (1942), 607–612. · Zbl 0061.14904
[16] 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
[17] Trjitzinsky, W. J.,Analytic theory of linear q-difference equations. Acta Math.61 (1933), 1–38. · Zbl 0007.21103
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