For a fixed $t$, I wish to find the value of,
$S = \sum_{k = -\infty}^{\infty} e^{ja_{0}^kt}$ where it is known that $a_{0} > 1$. I tried to express $S$ as a $z$ transform of some known function,
$S = \sum_{k = -\infty}^{\infty} 1 \cdot z^{a_{0}^k}$, where $z := e^{jt}$, but the exponent $a_{0}^{k}$ that appears on the $z$ term is causing a problem. Can anybody please help? Even if we could get a lower bound $f(t) \leq S$, it would be sufficient for my use. Thanks a lot.