To continue the analysis, let us recall some useful properties of the elementary symmetric functions:

Any alternant has a divisor--the Vandermonde's determinant of the same order, while the quotient can be uniquely expressed as a polynomial in elementary symmetric functions (32) (see Theorem 1).

The invariant algebra [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which consists of all the polynomials fixed by this 6n-action, is a polynomial algebra generated by the

elementary symmetric functions [e.sub.1], ..., [e.sub.n].

Parkhurst and James [79] tabulate zonal polynomials of order 1 through 12 in terms of sums of powers and in terms of

elementary symmetric functions.

To evaluate [e.sub.b][X + [M/z]] we will use the additivity of the elementary symmetric functions. That is, for any n and any expressions [E.sub.1] and [E.sub.2],

Furthermore, the elementary symmetric functions are homogeneous so

A series expansion of an elliptic or hyperelliptic integral in elementary symmetric functions is given, illustrated with numerical coefficients for terms through degree seven for the symmetric elliptic integral of the first kind.

The duplication method of computing the symmetric elliptic integrals [R.sub.F] and [R.sub.J] (including their degenerate cases [R.sub.C] and [R.sub.D]) consists in iterating their duplication theorems until their variables are nearly equal and then expanding in a series of elementary symmetric functions of the small differences between the variables.

We let [h.sub.i] and [e.sub.i] denote the complete homogeneous and

elementary symmetric functions of degree i respectively.

Their main examples of these Hopf-power chains were inverse shuffling (from the free associative algebra, with states indexed by its usual word basis) and rock-breaking (from the algebra of symmetric functions, with states indexed by the

elementary symmetric functions {[e.sub.[lambda]]}).

Our initial interest in the

elementary symmetric functions stems from the counting of degree n monic irreducible polynomials over finite fields with prescribed coefficients for [x.sup.n-1] and [x.sup.n-2].

are the generating functions of the homogeneous symmetric functions [h.sub.n] and the

elementary symmetric functions [e.sub.n] in infinitely many variables [x.sub.1], [x.sub.2],....