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.
Reviewer: Bogdan A. Choczewski (Kraków)
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 |
Keywords:
\(q\)-difference equation; zeros of solutions; entire functions; theta function; Newton-Puiseux diagram; geometric sequences; asymptotic relationsReferences:
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