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
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