Asymptotics for some fundamental \(q\)-functions (Q1936229)

While investigating the role of \(q\)-special functions in pure and applied mathematics, the derivation of asymptotic expansions is of great interest. Let us consider a real number \(q\in(0,1)\) and a complex variable \(z\). Then, the \(q\)-exponential \(e_{q}(z)\) and \(q\)-gamma functions \(\Gamma_{q}(z)\) can be defined as \[ e_{q}(x)=(z;q)_{\infty}=\prod_{n=0}^{\infty}(1-zq^{n}), \] \[ \frac{1}{\Gamma_{q}(z)}=\frac{\left(q^{z};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\left(1-q\right)^{z-1}, \] respectively. Also, in the case where \(n \in \mathbb{Z}\), the \(q\)-shifted factorial can be defined as \[ \left(z;q\right)_{n}=\frac{\left(z;q\right)_{\infty}}{\left(zq^{n};q\right)_{\infty}}. \] On the other hand, when \(n, m \in {0} \cup \mathbb{N}\), the Gaussian coefficient can be defined as \[ [\begin{smallmatrix} n\\k \end{smallmatrix} ]_q=\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{m}\left(q;q\right)_{n-m}}. \] Regarding the functional behavior, the study of the \(q\)-functions defined above when \(q \to 1^{-}\) is of interest. The development of asymptotic expansions in the latter limit is the central result of the paper. In order to derive the asymptotic expansions, the author considers a few inequalities. In every case, inequalities are proposed as lemmas and subsequently proved by means of standard series' manipulation techniques. The author presents the results in a clear way, which makes the text accessible to a readership with medium to advanced knowledge regarding series. Finally, it is important to point out that the asymptotic expansions obtained are independent of theta functions, which highlights the importance of the alternative approach employed.