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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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