Duplication formulae involving Jacobi theta functions and Gosper's \(q\)-trigonometric functions (Q2839310)

A new \(q\)-exponential function is defined as NEWLINE\[NEWLINE\exp_q(z):\frac2{\vartheta_2(q)}\sum\limits_{n=0}^\infty q^{\left(n+\frac12\right)^2}\exp\left((-1)^n(2n+1)\right).NEWLINE\]NEWLINE NEWLINEIt is motivated by the \(q\)-trigonometric definition introduced in [\textit{R. W. Gosper}, in: Symbolic computation, number theory, special functions, physics and combinatorics. Proceedings of the conference, 1999. Dordrecht: Kluwer Academic Publishers Dev. Math. 4, 79--105 (2001; Zbl 1058.33017)]. The author obtains the duplication formula NEWLINE\[NEWLINE\sin_q(2z)=q^{-\frac14}\frac{(q^2;q^4)_\infty^4}{(q;q^2)_\infty^4}\sin_{q^2}(z)\cos_{q^2}(z),NEWLINE\]NEWLINE where \((x;q)_\infty:=\lim\limits_{n\to\infty}(x,q)_n\). Finally, the author, using the above duplication formula, derives a new representation of the Jacobi theta function \(\vartheta_1\).