Is there a way to make sense of a bi-infinite generating function? For example, consider the series $$ \sum_{k=-\infty}^\infty t^k, $$ corresponding to the sequence $\dots,1,1,1,\dots$.
Is there a way to formally associate this series with a function of $t$ so that the usual generatingfunctionology applies?
I apologize if this sounds vague.
This is exactly the situation in Fourier analysis. In the simplest case you can think of this as the study of sums $$\sum_{k=-\infty}^{\infty} c_k z^k$$ where $z$ is restricted to the unit circle $(z=\exp 2\pi i \theta)$. You still have to worry about convergence issues; in particular the example you give $(c_k=1)$ doesn't converge.