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.

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