-

of interest tc consider, but if each element appears as a facter of
a term, i.e, with a numerical or‘literal coefficient, then it is of
interest tc know whether or not a permutation of the elements
affects the value of the function. If the number of elements is
finite, convergence dces not need to be considered. Since each
function of the infinite set of elements is an infinite series or
product, convergenceconditions and validity of mathematical opera-
tions are important tepics presented here, which, it seexs, have not
been discussed elsewhere. Moreover, the method of development per-
pits us to use the forrulas to transferm rany infinite products inte
infinite series. Examples of a number of different types are given,

the resulting series in some cases arparently being pew.?

Cefiniticns cf the elementary symmetric functions, the p's,

and the sums of pewers cf the elements, the s's.

If we have given a set of elements

BN W Ry iy e VeLiel g U e Tty

Tl B A Nl e N g it
() Y 448 GigEias T WU e 2
each row and each column bteing infinite in length, we can form the
fellowing functicns:
£y = S0y fos = Sb, foes = ¢4 Sy A A v e
$s = Sogay fea = Sheby Peos = E¢40; o P RE G RO ¥
Po ® 204040y fop =TDidsbr  foos =Teqesep tcc e e ees  waa.
$a1 = 20005 2403 = 30465 fors =TbiCi  fyan = Yojbye T

t21 = 30055k fg01 = 5a4040k topg = BOghych ) TR LT

e o o o o & S e e e e e ® 09 mi Al & “ e 0. 0.8 . .

¢ See especislly the Theta-series on page 8§.