Summing $\sum{\frac{1}{a^{P(n)}}}$ where $P$ is a given polynomial. If the sum: $$\sum_{n=1}^{\infty}\frac{1}{a^{P(n)}}$$ converges what is known about the value it converges to (if $a$ and $P(n)$ are known)? I am not sure if even the simple quadratic case can be solved.
 A: Well, for $P(n) = n^2$, 
$$  \sum_{n=1}^\infty \frac{1}{a^{n^2}} = \frac{1}{2} \left( \theta_3\left( 0 , \frac{1}{a} \right) -1 \right)  \text{,}  $$
where $\theta_3$ is the third Jacobi theta function.  (See (10) at MathWorld: Jacobi Theta Functions.)  It's only convergent for $\frac{1}{a}$ in the (complex) unit disk.
In the following, I have assumed that $P(n) \in \mathbb{Z}[n]$, that is the set of polynomials in $n$ with integer coefficients.  If the intened polynomials take non-integer values for $n \in \mathbb{Z}$, this question becomes much more complicated.  (See Robert Israel's comment for a sampler.)
I don't have a reference, but I suspect "many" quadratic $P(n)$ can be handled by a little ingenuity with the four theta functions.
Regarding convergence ...
Since $P(n)$ of degree $\geq 2$ eventually grows aperiodically, it is not rational, so has the unit circle as a natural boundary.  See Bell et al. (Bell, Jason P., Nils Bruin, and Michael Coons, "Transcendence of Generating Functions Whose Coefficients are Multiplicative", https://arxiv.org/pdf/1003.2221.pdf ) to get a start into the literature.  Their Theorem 1.2:

Theorem 1.2 (Carlson [10]).  A series $F(z) = \sum_{n \geq 1} f(n) z^n \in \mathbb{Z}[[z]]$ that converges inside the unit disk is either rational or it admits the unit circle as a natural boundary.

In your variables, $z = 1/a$, and $f(n) = \begin{cases}1, &n \in \{P(k) \mid k \in \mathbb{Z}_{>0}\} \\ 0, &\text{otherwise}  \end{cases}$.  A rational function eventually has periodic $f$, but the gaps between values of $P(k)$ grow without bound.  Consequently, if $\deg P \geq 2$, your series only converges for $1/a$ on the unit disk.  That is, it only converges for $a$ outside the unit disk.  It can't even be analytically continued into the disk.
