gn.general topology - Euler characteristics and operator indices as exponents for Laurent polynomials

This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$sum q^{a_i}inmathbb{Z}[q,q^{-1}]$$
where the $a_i$'s are either Euler characteristics of some spaces (possibly all subspaces of one fixed space), or more generally, indices of some elliptic operators? I've stumbled across such a beast, but am unsure how to interpret it. I was thinking at first that it was an element of $K_{S^1} (pt)$ or something, but in that case the exponents are telling us about which $S^1$ representations show up in the appropriate bundles, not the indices of the operators, right? Is there some relation with the index? (Please tell me if I'm talking nonsense! I don't really know this K-theory stuff).

Maybe the answer I'm looking for doesn't involve K-theory, anyway. Does anyone have any ideas? I'd love to hear about any and everything!