Bounds of Summation - What happens when the upper bound is below the lower bound

Question

I have the following proposition:

$$ P(n) : (x^n - y^n) = (x-y) \sum_{i=0}^{n-1} x^{n-1-i} y^i $$

If the proposition is tested for $n=0$, the upper bound of the summation is lower than the lower bound of the summation:

$$ P(0) : (x^0 - y^0) = (x-y) \sum_{i=0}^{-1} x^{-1-i} y^i $$

I'm coming to Mathematics from Software Engineering; in Software Engineering, we would conclude that $n$ must be limited to Integers greater than or equal to one, or the Natural Numbers.

How is this usually handled in formal mathematics? The same way--by explicitly specifying the domain of $n$ as $\mathbb{Z} \setminus \{ 0 \}$ or $\mathbb{N}$?

Are indices on iterative operators assumed to have domains that are in $\mathbb{N}$ by default in formal mathematics?

I'm aware that in Calculus, integration, the bounds of integration can be flipped so that the lower bound is lower than the upper bound.

...I kind of feel like I'm answering my own question here as I type:

• In mathematics, the bounds of the $\sum$ operator are assumed to be in $\mathbb{N}$, unless explicitly stated otherwise.

• In integration, the bounds of integration are usually in $\mathbb{R}$ or the complex plane.

• With any iterative operator like $\sum$, $\prod$, etc. if it has been proven that the bounds of operation can be safely flipped, it is okay to flip bounds. Otherwise, one must explicitly state the domain of one or more factors in the bounds of operation to ensure the upper bound is greater than the lower bound.

Is that correct? How's that for jargon gibberish?

Answer 1

Here are a few aspects about this theme from section 2.1 Notation of Concrete Mathematics by D.E. Knuth etal.

• In general we consider the symbol \begin{align*} \sum_{P(k)}a_k \end{align*} as an abbreviation for the sum of all terms $a_k$ such that $k$ is an integer satifying a given property $P(k)$. A property "$P(k)$" is any statement about $k$ that can either be true or false.

• If $P(k)$ is false for all integers $k$, we have an empty sum.

• The value of an empty sum is defined to be zero.

In case of \begin{align*} \sum_{k=0}^{-1}a_k \end{align*} the index range is empty. So, we have an empty sum and it's value is defined to be zero.