TITLE: Identifying the generating function $ G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}. $
QUESTION [10 upvotes]: I have computed a generating function for a problem involving a particular series, and would like to know if anyone has any references or a categorisation for it? It's
$$
G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}.
$$
It appears to be related to (mock) theta functions, but seems to be simpler.
In particular, I would like to know whether $G(a,z)$ satisfies any identities?
Many thanks.

REPLY [5 votes]: Your generating function is related to a simple continued fraction expansion due to Touchard: 
$\sum\limits_{k \ge 0} ( - 1)^k q^{k+1\choose2} v^k $ =$ \frac{1}{{1 + v - \frac{{(1 - q)v}}{{1 + v - \frac{{(1 - q^2 )v}}{ \cdots }}}}}.$
A simple proof can be found in a paper by H. Prodinger
  http://de.arxiv.org/abs/1102.5186