Another addition theorem for the \(q\)-exponential function (Q2711832)

For the basic exponential function defined by NEWLINE\[NEWLINE\mathcal{E}_q(x,y;\alpha)=\frac{(\alpha^2;q^2)_{\infty}}{(q\alpha^2;q^2)_{\infty}} \sum_{n=0}^{\infty}\frac{q^{n^2/4}\alpha^n}{(q;q)_n}e^{-in\varphi} (-q^{(1-n)/2}e^{i(\varphi+\theta)},-q^{(1-n)/2}e^{i(\varphi-\theta)};q)_nNEWLINE\]NEWLINE with \(x=\cos\theta\), \(y=\cos\varphi\) and \(|\alpha|<1\) it can be shown that \(\mathcal{E}_q(x,0;\alpha)\mathcal{E}_q(y,0;\alpha)=\mathcal{E}_q(x,y;\alpha)\), which can be seen as a (commutative) \(q\)-analogue of the addition theorem \(\exp(\alpha x)\exp(\beta x)=\exp(\alpha(x+y))\) for the ordinary exponential function. A proof can be found in [\textit{S. K. Suslov}, Methods Appl. Anal. 4, No. 1, 11-32 (1997; Zbl 0877.33008)] for instance. Although \(\mathcal{E}_q(x,0;\alpha)\) is a \(q\)-analogue of \(\exp(\alpha x)\), the function \(\mathcal{E}_q(x,0;\alpha)\) is not symmetric in \(x\) and \(\alpha\). In the paper under review the author proves a formula for the product \(\mathcal{E}_q(x,0;\alpha)\mathcal{E}_q(y,0;\beta)\) which generalizes the above mentioned formula. The result can be seen as a \(q\)-analogue of the addition theorem \(\exp(\alpha x)\exp(\beta y)=\exp(\alpha x+\beta y)\). The limiting cases \(\alpha\rightarrow 0\) and \(\beta\rightarrow 0\) lead to generating functions for the continuous \(q\)-Hermite polynomials in terms of the above mentioned \(q\)-exponential function.