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.



Identifying the generating function $ G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}. $
It should be noted that using the Jacobi triple product that we have
$$
H(a,z) = a^{-1}\sum\_{n=-\infty}^\infty a^nz^{n(n+1)/2}
= a^{-1}\prod\_{m=1}^\infty (1 - z^m)(1 - z^ma)(1 + z^{m-1}a)
$$
where the main difference is that the indexing shifts and we are doing about twice the sum that you are.


If we try relate these, we get
$$
H(a,z) = \big(a^{-1} + G(a,z)\big) + a^{-2}\big(1 + a^{-1}G(a^{-1},z)\big).
$$


I'm not sure how much this helps though.


