Relations between coefficients of expansions of a rational function at 0 and infinity

Question

This question goes in the bucket of "this must be well known, but I don't see it and am not sure where to look it up."

Given two Laurent power series $A(t)=\sum_{k>N}a_kt^k$ and $B(t)=\sum_{k>M}b_kt^k$ for $a_k,b_k\in F$ a field, we say that these are expansions at 0 and $\infty$ of a rational function $f$ if in the field $F((t))$, we have that $f(t)=A(t)$ and $f(t^{-1})=B(t)$.

Now, think about $\,a_k$ and $\,b_k$ as formal variables:

Are there any relations between $a_k$ and $b_k$, or any other way of expressing algebraically that $A(t) = B(t^{-1})$?

It seems to me that there can be no relations in the strong sense that I can fix any finite number of $a_*$ and $b_*$ and complete to get such compatible expansions, but maybe there's some more subtle relation I've missed.

EDIT: After feeding the answer below, I see what I should have thought of before: this is actually a topological property. We can endow the ring generated by $a_k$ and $b_k$ with a topology such that the ideal generated by the $m\times m$ minors of the Hankel matrix for each $m$ give a basis of neighborhoods of $0$. A map of this ring to the base field $F$ (with the discrete topology) is continuous if and only if one of these neighborhoods is killed, and we have a rational function that we have the two expansions of.

Answer 1

One common criterion that is used to algebraically encode whether a power series $\sum_{n\geq 0} a_nx^n$ represents a rational function is that the infinite Hankel matrix $(a_{i+j-1})$ be of finite rank. If this is the kind of criterion you are looking for then for your pair of series $A(t)$ and $B(t)$ satisfy $A(t)=B(t^{-1})$ if and only if the (bi infinite) Hankel matrix associated to $$C(t)=A(t)-B(t^{-1})$$ is of finite rank. That is, express $A(t)-B(t^{-1})$ as a power series $\sum_{n\in \mathbb Z}c_n x^n$ and then check that the bi infinite matrix $(c_{i+j-1})$ has finite rank. This is the same as saying that certain determinants with entries of the form $a_k-b_{-k}$ vanish.

More explicitly this is encoding the fact that if, for example, $A(t)$ looks like $$\sum_{n\geq 0}\left(P_1(n)\alpha_{1}^n+\cdots P_k(n)\alpha_k^n\right)t^n$$ then $B(t)$ looks like $$-\sum_{n\geq 0}\left(P_1(-n)\alpha_{1}^{-n}+\cdots P_k(-n)\alpha_k^{-n}\right)t^n.$$ In other words as formal power series we have a "vanishing relation" $$\sum_{n\in \mathbb Z} P(n)\alpha^nz^n=0$$ This last relation is a nonrigorous way of expressing what I said above, but I like thinking about it as a relation in the same sense as in Brion's formula from polyhedral generating functions/equivariant localization (and it can be made rigorous using the same methods).