Based on the vagueness of the power term too, I suggest further that the
definition be replaced with "generalized Laurent 'polynomial' function".
In this situation, one must specify the two degrees (nonnegative and
nonpositive over which the summation is occurring); the input domain of
the function and the steps of summation (is it natural numbers? What if
the degree is √2?) should be specified in the definition too. The good
news is that we now would have a very general function/mathematical tool
that has a robustly defined name (outside of English usages if
"polynomial" but simple generalizations really) and which is exact and
clearly different from "formal polynomial" (which is not the case in
English).

I think that this word is a good one, but the definition needs to be
mathematically more exact and correct.