Let $F_0=0,F_1=1,...$ be the Fibonacci numbers. Is there a known closed form for the sum $\sum\limits_{n=0}^\infty q^{F_n}$? By closed form, I mean in terms of well-known functions, the first ones to think of would probably be theta functions.
The fact that $\sum\limits_{n=-\infty}^\infty q^{F_n}$ diverges everywhere makes me rather pessimistic, but maybe this series has been studied somewhere?