$f_C(e^{2iz})+f_C(e^{-2iz})=1+\sum_{k=-\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$ for $u=C$ it's an elliptic theta function,

$$f_u(u)=1+\frac{1}{2}u^{1/4}\vartheta_2(0,1/u)$$

more generally $$f_C(e^{2iz})+f_C(e^{-2iz})=1+\sum_{k=-\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$$

$\vartheta_3(z,q)=1+2\sum_{n=1}^\infty q^{n^2}\cos(2nz)$$f_C(e^{2iz})+f_C(e^{-2iz})=1+\sum_{k=-\infty}^{\infty}\frac{e^{2kiz}}{C^{k^2}}=1+\vartheta_3(z,1/C)$