-4

20~

If we write out a set of equations like (12), for the
elepents of (1'), but where each £' and s' is written out in terms
of its elements according to the definitions cn pages 2 and 3, each
equation is of the form C, = 9(L,¢) = ¥(¥,s), which is an iden-
tity in the Y's, and ¢ and y are both pclynomials in the Y's
(because jo+ i + ****+rA=w has a finite number of solutions).
This identity holds for C £ ¥ £ 1, since #' and s' are absolutely
convergent when I = 1, Freom formula (3) and the relation of ' teo #
we see that every term of ¢(1,#) has the form YJRF e eh” s, s
where jo+ 8 + ****+rd=w. Fcrmula (11) shows that in w(7,s) there
are teros of the forw A(Yahlek’“'if’ s]v:h...,i)"l"'(Yaj‘“'n\r‘ s“,.r‘)"‘
where A depends on the indices, and where the indices satisfy the
equaticn Zmihy = Imi(jia + kB 4 v 4 rA) = Grgida 4 Crk)S 4+ vz w,
Since we must use all values of the indices which satisfy this con-
dition, there is at least one set of pumbers such that Zn,j; = j,
EMgky = k, °°°*°, Tmyry = r, where j, k, ***°, r, are those given
with a certain pjke..r . Therefore there is in y(Y,s) a term having

Y“’YB“""YA’ with a coefficient expressed in terms of s's, for each

YO‘JYB”'"'Y;\' Pikeeeer - existing in o(Y,p) . Since the polynomial ¢

equals the peclynomial y identically in the Y's, the coefficient of
a power of the Y's in ¢ equals identically the coefficient of the
sage power of the Y's in y.® Therefore each ¢ may be expressed in
terms of s's, since each p occurs just once in ¢, and we have
proved the fellowing

Turorex V. ' The function Pjp,..r can be extressed in terms

of the s's by a formula derived frow equatians (12), via.

(13) (_1)‘2‘“";' R ey 3 :
Bikicop TRATY A A Yitewve(s )t

JReove b "" J,""i-" J-”"‘ J,k,..."; “k,...r‘ ’
1°° o Jyea §oe

e e e A A es e
Bocher, sec,%, definition on page 5, and Theorem 2, page 6.
Also Niewengloski, p, 12.

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