Notation for writing down products of sets of combinations

Question

I am writing a paper in which I come across an expression analagous to;

$$ \prod_{k=0}^{n} (x-r_k) $$

I wanted a nice way of writing down how the $r_n$ relate to the coefficients in the resulting polynomial. These coefficients are the sums of products of the set of all combinations (where the length of the product can be related to the power of the $x$ it sits next to.

As an example, consider $$(x-a)(x-b)(x-c) = -abc + (ab +ac+bc)x - (a+b+c)x^2 +x^3$$

You can see the coefficients could be (stupidly) written as

-(the sum of all combinations of products of length 3) + (the sum of all combinations of products of length 2) - (the sum of all combinations of products of length 1) + 1

I'm looking for a nicer way of writing it.

I'm sure there must be a notation for this, but I'm a Physicist, so I haven't seen it!

Answer 1

Your coefficients are the elementary symmetric polynomials in the roots. Any notation for such polynomials will do, I guess.

Answer 2

Here's a nifty notation.

For any set $S$ and any natural number $n \ge 0$, let $\binom{S}{n}$ denote the set of all $n$-length combinations on $S$. For example,

$$\binom{\{a, b, c\}}{2} = \{\{a, b\}, \{a, c\}, \{b, c\}\}$$

Let $R$ be the set of roots, so that $R = \{r_0, r_1, r_2, \ldots, r_n\}$. You can express the product as follows:

$$\prod_{r \in R} \left(x - r\right) = \sum_{i = 0}^{n+1} (-1)^i \left( \sum_{R' \in \binom{R}{i}} \left( \prod R' \right) \right)x^{n+1-i}$$

Note the various ways to express a sum or product!

Though if you prefer a more conventional style that uses increasing powers of $x$, you can opt for:

$$\prod_{k = 0}^{n} \left(x - r_k\right) = \sum_{i = 0}^{n+1} (-1)^{n+1-i} \left( \sum_{R' \in \binom{R}{n+1-i}} \left( \prod_{r \in R'} r \right) \right)x^i$$