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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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