Fart I
Thecry of Syrmetric Functions

Introduction

A fundamental problem of symmetric functions is that of
expressing a symmetric I function of one kind in terms of symme t-
ric ¥ functions of another kind. The already published sclutions
of the prcblem seem to be valid only when the number of elements
and of sequences is finite, and those scluticns require a step by
step process of building up from the simplest functions, except
when there is just one seauence of elements.® The solution given
here is valid for an infinity of sequences, each compesed cf an

infinity of elements. A step by step Frecess is not required, and
all the finite cases are included as simple special cases. Ry
transforring intc series an infinite product each factor of which
is an infinite series, formulas are derived for expressing symmet—
ric functions of cne kind directly in terms ¢f ancther kind. The
heretofcre difficult and tedicus problem of expressing any general
sygmetric 3 functicn in terms of the elementary functions cr the
surs of rowers of the elements is done here by reans ¢f a few sim~
ple substitutions into the formulas. The various finite cases are
obtained by assigning the value zero tc all the elements not in
the given finite set. In this entire paper no special consideratjon
will be given to a finite number cf elements or sequences, except
when examples are given tc show that such are special cases.

In a polygomial or infinite series which is a symmetric
functicn, if the whcle terws are the elements, there is ncthing

e el st i S S 0 3 SR TN B U

! See Cullis or Junker for the step by ster rrocesses. Iresden #ives a formulas
for the problem when there is only one seguence,

The complete titles of all references are listed at the end of the paper,

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