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