Wiman-Valiron theory for a polynomial series based on the Askey-Wilson operator. (English) Zbl 1477.30021
Summary: We establish a Wiman-Valiron theory of a polynomial series based on the Askey-Wilson operator \(\mathcal{D}_q\), where \(q\in (0,1)\). For an entire function \(f\) of log-order smaller than \(2\), this theory includes (i) an estimate which shows that \(f\) behaves locally like a polynomial consisting of the terms near the maximal term of its Askey-Wilson series expansion, and (ii) an estimate of \({\mathcal{D}}_q^n f\) compared to \(f\). We then apply this theory in studying the growth of entire solutions to difference equations involving the Askey-Wilson operator.
MSC:
| 30D10 | Representations of entire functions of one complex variable by series and integrals |
| 30B50 | Dirichlet series, exponential series and other series in one complex variable |
| 30D20 | Entire functions of one complex variable (general theory) |
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