"requires" is a strong word. the only explicit definition ive seen given for n-ary summations and products products was recursive.

I don't think so. For example, sum of first 5 integers:

    sum i = 1 to 5 of i -> 1 + 2 + 3 + 4 + 5 = 15

Nothing mutable there. In functional programming this is done with a `fold`, which doesn't use mutation.

We're talking about notation. Anything could be described in any Turing complete language. It's uncommon to have something similar to mutation in ordinary mathematical notation, but summation is one example where it's used.