## Zeros of solutions of a functional equation.(English)Zbl 1087.39022

The $$q$$-difference equation $\sum_{j=0}^m {a_j(x)~f(c^jx)} = Q(x)\tag{1}$ is considered in the class of transcendental entire functions. Here $$c \in {\mathbb C},~0 < |c| < 1$$, and $$Q$$ and the $$a_j$$ are polynomials. Imposing conditions on the associated Newton-Puiseux diagram the authors show a theorem saying that (sub)sequences of zeros of solutions $$f$$ to (1) are asymptotically comparable with certain geometric sequences. The proof is achieved by showing that $$f$$ behaves asymptotically like a product of theta functions, $$\theta(z,q) = \sum_{n = - \infty} ^{+ \infty} {q^{n^2}z^n}$$. The results are illustrated on three examples, the latter showing that the conditions imposed in the theorem on the Newton-Puiseux diagram are basically essential.

### MSC:

 39A13 Difference equations, scaling ($$q$$-differences) 39B32 Functional equations for complex functions 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 33D90 Applications of basic hypergeometric functions
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### References:

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