Newton Sums

Let be variables. In this post we develop the Newton formulas relating the symmetric polynomials in the

and the power sums

(For example, when we have 3 variables, and . For convenience of notation we don’t restrict the number of variables; if we are working with variables we could just set . By convention .)

We note that the and are the coefficients of powers of of the following generating functions.

Why do these expansions hold? Each term in the expansion of is a term of the form where the are distinct, since we can only get a factor of from the factor . Since each always comes with a factor of , acts as a “counter” giving the total number of variables. Grouping the terms with we get all combinations of products of of the .

For the argument is simpler: expand in geometric series as shown above to get that the coefficients of are the th powers of the .

We want an identity involving and so we look for an identity involving and . First we turn the into : define

Now take the logs of both sides (thinking of the above as a formal series in ):

Now differentiate both sides.

The right-hand side is just (TA-DA!). Thus we get

Now we match coefficients of on both sides. Note

so the coefficient of is . To get the coefficients of the right-hand side, note that a term containing on the RHS comes from multiplying in and a term in . Thus the coefficients of are

which is the Newton sum formula.