Are "sum" and "product" defined when there is only one number?

You would probably agree that $$ \sum\limits_{n=1}^Nx_n=x_N+\sum\limits_{n=1}^{N-1}x_n\quad\text{and}\quad \prod\limits_{n=1}^Nx_n=x_N\cdot\prod\limits_{n=1}^{N-1}x_n. $$ Try $N=2$ in these formulas and identify $$ \sum\limits_{n=1}^{1}x_n\quad\text{and}\quad\prod\limits_{n=1}^{1}x_n. $$ By the same argument, any sum of zero terms is $0$ and any product of zero terms is $1$.

The short answer is "Yes".

We can define multiplication $a\cdot b$ as an $a$ long sum of $b$'s. Think of objects alligned in a rectangle, the way you're taught in elementary school. What then would the product $1\cdot5$ mean? It would be the total sum of one $5$. For multiplication, the same argument can be made with $5^1$. But it gets better.

What is the sum of no numbers? That's easy to calculate, $0\cdot b= 0$. But how do you make sense of that? Imagine having a box full of numbers, and a display on the lid of the box what the sum of those numbers are. Then take them out, one by one, and see how the display changes. The intuitive consequence is that when you remove the last number from the box, the display should read "0".

What then, of the product of 0 numbers? Imagine a calculator with a number pad, a clear-button, an enter-button and a one-line display. It has one number stored, and you can type in another number. As long as you're not typing, the stored number is shown on the display. When you press the "enter"-button, the calculator multiplies the typed number wiht the stored number, stores the result, and displays it. When you press "clear", you would intuitively have a product of no numbers stored, and the only number to store that makes the calculator work is 1. So the empty product is 1. You see it in math as for instance $5^0$, and $0!$.