## 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:

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