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.
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