-10-

has a meaning for all choices of the weights a, 8, y, ==<-, A, oe-
another notation will be used. Denote f£4p...q by ¢#[jk°c*q] . Let
r, s, ***, t, be chosen so that r/a = j, s/B =k, ****°, 4, = G
In this notation pjpee.q = #[ Eg ""% 1.. This puts the weights
in evidence in the symbol. For any choice of the integers a, B, °--
A, ***, r, s, ***, t, the symbol has no meaning if any one of the
numbers r/a, ----, t/\, does not reduce to an integer, Also, the
weight is jo+ M+ **** 4 h=Lus 39+ v tfh=rase i,
Writing the C's in terms of the #'s,

Cy = f[é] ’
Co= $[21 + (23],
(3" e,= spl5g e E1

where r/a, s/B, *-°- t,/\, are integers and r+ s4 **** 4 t=n,

TazoreM II. Given the array (1) such that the f's and v's
converge, and vy(x) 5 =1, and 3|vy(x)| converges uniforwly. Then

(2) retresents II, where the C's are given by (3) or (3'),

We shall next transform the product I = II (1 + v;(x)]
into a power series 1+ Cyx 4 Cox® 4 ***** 4 Cox™ 4 **** | obtaining the
formula for C, expressed in terms of the s's. We wish to find the
least restrictive conditions under which the transformation can be
performed; for this reason no hypothesis sball be stated at the
teginning and no condition shall be irposed until needed.

We shall use the following method. First we shall define

- log I, so that w=73 log[l + vyl =3 [vi=3v2 4 §o® eeeee ],
Each power of v; is a function of x, and after we find these powers
of vy we shall arrange log[l + v4] according to powers of x, Ob=-

taining log([l + vy(x)] = %unxh . Summing with respect to i we can

i h y g
obtain ‘E%"ih" =u, but by interchanging the order of summation

we obtain u expressed as a power series in which the coefficients

o