Asymptotics of zeros of basic sine and cosine functions (Q1873785)

The \(q\)-sine and \(q\)-cosine functions studied in this article are \[ \begin{aligned} S_q(\eta;\omega) &= \frac{1}{(-q\omega^2;q^2)_\infty} \sum_{k=0}^\infty \frac{(-1)^kq^{k(k+1/2)}}{(q^{1/2};q^{1/2})_{2k+1}}\omega^{2k+1}, \\ C_q(\eta;\omega) &= \frac{1}{(-q\omega^2;q^2)_\infty} \sum_{k=0}^\infty \frac{(-1)^kq^{k(k-1/2)}}{(q^{1/2};q^{1/2})_{2k}}\omega^{2k}. \end{aligned}\tag{*} \] For the purposes of this article, the parameter \(\eta\) plays no role and could be left out here, but the author prefers to retain it as a reminder that these functions are both special cases of more general functions. By taking the limit termwise, we see that \[ \lim_{q\rightarrow 1} S_q(\eta,\omega(1-q^{1/2})) = \sin \omega, \] and the cosine arises similarly as a limiting case of the function \(C_q(\eta;\omega)\). When \(0<q<1\), it is known that all zeros of \(S_q\) and \(C_q\) are real and simple, and the positive zeros of \(S_q\) and \(C_q\) interlace. In this article, asymptotic formulas for the \(n\)-th positive zeros of \(S_q\) and \(C_q\) are proved. These results are stronger than conjectures made by \textit{R. W. Gosper} and \textit{S. K. Suslov} [\(q\)-series from a contemporary perspective. AMS-IMS-SIAM joint summer research conference on \(q\)-series, combinatorics, and computer algebra, Contemp. Math. 254, 199-227 (2000; Zbl 0956.33010)]. As an application, it is shown that the asymptotic formulas imply that a \(q\)-analogue of the Riemann zeta function has simple poles at \(z=1, -1\) and \(-2\), and no other singularities in the half-plane Re\((z)>-3\). Analytic continuation of this \(q\)-zeta function to the entire complex plane evidently remains an open question.