Background
An Egyptian fraction is a finite sum of distinct unit fractions, in which each denominator is not bigger than the next one. In other words, it is a representation of $a/b$ such that $$\frac{a}{b} = \sum_{n=1}^{N} \frac{1}{c_{n}} $$ with $c_{1} \leq c_{2} \leq c_{3} \leq \dots c_{N} .$ For instance, we have $$ \frac{80}{99} = \frac{1}{2} + \frac{1}{4} + \frac{1}{18} + \frac{1}{396} .$$
I'm currently exploring the idea of an Egyptian product (*). It is a representation of a fraction $a/b$ with $b>a$ such that $$\frac{a}{b} = \prod_{k=1}^{K} \left(1-\frac{1}{a_{k}} \right) ,$$ where $a_{1} \leq a_{2} \leq a_{3} \leq \dots \leq a_{K} .$ If we take the previous fraction again as an example, we find: $$ \frac{80}{99} = \left(1-\frac{1}{9} \right) \left( 1-\frac{1}{11} \right). $$
Questions
- Has this notion of an Egyptian product been described in the mathematical literature before?
- Does every fraction $0 < \frac{a}{b} < 1 $ have an Egyptian product representation?
- Is the Egyptian product a new and "interesting" representation in the sense that they can't be reduced to finding Egyptian fraction sums (or perhaps a particular type thereof) somehow?
Notes
(*) Sometimes, a particular type of Egyptian fraction - the Engel expansion - is also called an Egyptian product. The concept of an Egyptian product I have in mind is different.
The answer to question 2 is yes: for any positive integers $m$ and $n$, $$\frac m{m+n}=\prod_{k=1}^n\left(1-\frac1{m+k}\right).$$