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.

中文翻译：

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基于Askey-Wilson算子的多项式级数的Wiman-Valiron理论

我们基于Askey-Wilson运算符\（{\ mathcal {D}} _ q \）建立一个多项式级数的Wiman-Valiron理论，其中\（q \ in（0,1）\）。对于对数小于2的整个函数f，此理论包括（i）一个估计，该估计表明f的局部行为类似于多项式，该多项式由接近其Askey-Wilson级数展开的最大项的项组成，和（ii）相对于f的\（{\ mathcal {D}} _ q ^ nf \）的估计。然后，我们将这一理论应用于研究涉及Askey-Wilson算子的差分方程的整体解的增长。