Pi product notation

Question

The exact expression I've seen in a paper looks like this:

$$\displaystyle \prod_{k<l}^L(x_k-x_l)$$ where $L$ is some natural number. What does the product actually look like when expanded out?

Answer 1

It is the product of all such factors where the first term has index less than the second and both terms have index less than or equal to $L$. In other words

$(x_1 - x_2)(x_1 - x_3) \cdots (x_1 - x_L)(x_2 - x_3)(x_2 - x_4) \cdots (x_2 - x_L)(x_3 - x_4)(x_3 - x_5) \cdots (x_3 - x_L) \cdots (x_{L-1} - x_L)$

Answer 2

It is a notation that can (unfortunately) mean one out of two things (or more). The first posibillity is the product over all $l$ s.t. $k<l\leq L$. This product is

$$(x_k-x_{k+1})(x_k-x_{k+2})\ldots (x_k-x_{L})$$

The second possibillity is that it is the product over all integers $k$ and $l$ s.t. $1 \leq k<l \leq L$ , i.e.

$$[(x_1-x_{2})(x_1-x_{3})\ldots (x_1-x_{L})] \times [(x_2-x_{3})(x_2-x_{4})\ldots (x_2-x_{L})] \times \cdots [(x_{L-1}-x_{L})]$$

The meaning is usually interpreted from the context the expression is given in (e.g. if it is written something like $a_k = \prod_{k<l}^L(x_k-x_l)$ then it is clearly the first option)

There are also many other variations here: in some cases the product starts at $k=0$ instead of $k=1$, and again the meaning is usually to be infered from the context. The thing to take from this is that one should always try to use a notation that leaved no room for speculation (like for example $\prod_{l=2}^L\prod_{k=1}^{l-1}(x_k-x_l)$ for the second option above)