The elementary symmetrical polynomials (ESP's) are polynomials in n indeterminates that are symmetrical, that is, they don't change under the permuation of the indeterminates.

They are defined as follows.

S_{1}(t_{1}, t_{2}, , t_{n}) = t_{1} + t_{2} + + t_{n}

S_{2}(t_{1}, t_{2}, , t_{n}) = t_{1}t_{2} + t_{1}t_{3} + + t_{1}t_{n} + t_{2}t_{3} + t_{2}t_{4} + + t_{2}t_{n} + t_{3}t_{4} + + t_{n-1}t_{n}

S_{3}(t_{1}, t_{2}, , t_{n}) = t_{1}t_{2}t_{3} + t_{1}t_{2}t_{4} + + t_{1}t_{2}t_{n} + + t_{1}t_{n-1}t_{n} + + t_{n-2}t_{n-1}t_{n}

S_{n}(t_{1}, t_{2}, , t_{n}) = t_{1}t_{2} t_{n}

In other words, S_{k} exists of a sum of all permutations of products of precisely k different t_{j}. There are such permutations for a given n and k.

A more formal way of defining the ESP's is

The Newton Symmetric Polynomials (NSP's) on (t_{1} t_{2} t_{n}) are the polynomials

For a fixed n, the set { S_{1}, S_{2}, ..., S_{n} } is linearly independend; they cannot be expressed into eachother. Since there are n indeterminates and n elementary symmetric polynomials, they form a basis for the set of symmetric polynomials of n indeterminates. The same holds for the set { N_{1}, N_{2}, ..., N_{n} }. Every symmetric polynomial can be expressed as a polynomial of the ESP's or NSP's, in particular they can be expressed into eachother.

A recursive formula that expresses one into the other still looks reasonably simple. Therefore this is the prefered way. For a fixed n the relationship is given by

from which it is easy to derive the reverse relationship

Both cases exist of a sum with k - 1 leading terms, followed by the last term. In particular, N_{1} = S_{1}.