Abstract
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.
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Notes
The logarithmic density of a set \(E\subseteq [1,\infty )\) is defined by
$$\begin{aligned}\mathrm {logdens}\,E:=\limsup _{r\rightarrow \infty }{\frac{\mathrm {logmeas}\,(E\cap [1,r])}{\ln r}}=\limsup _{r\rightarrow \infty }{\frac{1}{\ln r}\int _{E\cap [1,r]}{\frac{1}{x}\,dm}}, \end{aligned}$$where m is the Lebesgue measure on \({\mathbb {R}}\).
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The authors would like to thank the anonymous referee for her/his helpful and constructive comments and bringing Fenton’s work [12] to their attention.
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Communicated by Mourad Ismail.
Dedicated to the memories of Jim Clunie and Walter Hayman.
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Both authors were partially supported by GRF No. 16306315 from the Research Grant Council of Hong Kong. The first author was also partially supported by the PDFS (No. PDFS2021-6S04), and the second author was also partially supported by GRF No. 600609 from the Research Grant Council of Hong Kong.
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Cheng, K.H., Chiang, YM. Wiman–Valiron Theory for a Polynomial Series Based on the Askey–Wilson Operator. Constr Approx 54, 259–294 (2021). https://doi.org/10.1007/s00365-021-09528-3
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DOI: https://doi.org/10.1007/s00365-021-09528-3