help with the closed form for a potential cons seq

[Marc LeBrun]

 > Can you close Sum[2^(-k*(k + 9)/9), {k, 0, Infinity}]?
 > Or more usefully/generally, Sum[2^(p*k^2 + q*k + r), {k, 0, 
Infinity}] for p,q,r in the rationals and p negative?
 > I note the specific value is approx'ly 1.733126470890002...

It depends on what you mean by "closed".

Sums like this (with quadratic exponents) are often expressed via the 
"Jacobi" elliptic theta functions.

http://mathworld.wolfram.com/JacobiThetaFunctions.html

Since they satisfy many identities, it might happen that you could 
express the value you seek at the point x=2 in some more familiar or 
elementary terms, but that would be unusual.

Of course these functions are of great interest in their own right, 
for many reasons, and might reasonably be considered as "closed" a 
form as, say, an expression involving sin(2).