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.



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


