What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it.
So, what do you think the summation actually is when expanded?
An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w.
You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.
0^0 = 0
1^0 = 1
0^1 = 1
1^1 = 0
exp(x) = summation(x^n/n!, n=0, inf)
Fair, infinite series are pretty esoteric. How about derivatives. The power rule:
d/dx x^n = n x^(n-1)
Eh maybe not, programmers don't use calculus that often. But surely statistics!
(1 + x)^n = summation((n choose k)x^k, k=0, n)
(1 + x)^n = (1 + 0)^0 = 1 = 0^0 = x^0 = x^0 + ... + x^n
When you see something like.
We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.