-22~
therefore the nth power converges absclutely,® and can be arranged w®:
according to powers of x. In order to have u converge absolutely
and uniformly, we must determine Ry < Ry such that | -1 <1-c¢,

(0O<€<1), when |x| <R, ., From the multinomial theorem
61" (-1)" =

e (Cyx + CQZ:"’ e )1& =Y T

 

(C1x¥ (Cax®)® 420 (Cx*)F

=3 ‘(:l(_)]%_:i zdo.n)»...ur Cz’ C'h e g,
where the summation is taken for the values of the indices such
that j+k+ ***+r=n (=m+1). Now each term of w §s absolutely
convergent and the series for u converges uniformly. Therefore, by
the Weierstrass theorem, the series can be arranged according to
powers of x, Picking out all the terms having a given power, k, of

x, we have

(14) Uy = Zs:l—).'.—:..c;’ C:h “ove C"'

’

where j+2k+ *ectir=h, m=j4+k+ " 4r-1 and v =2uh %

Using (14) u=Cax + x%(= 4C4% + €3) + x®( $C1® = €103 4 Cs) + **°+ ,

Using (7) u=sx+ x°(=§sg + s03) + x°( sa = 833 + 3002) + ‘v,
if we take « =1, 8 = 2, y =3, *»-~- _ Since each of these saories
is a power sgries representing the same function for all values of
|x] < Ry, the coefficients of the same powers in the two series are
identically equal.®*

TaEoREN VI, If the elements of the array (1) satisfy the

hypothesis of Thearew 1V, then for each fixed integer h,

 

)" g GO =t
z T Sdkeser = —‘7,—"‘" Cy? Co¥ eove
o .

shere on the left jo+ B + ***+vrh=h, and on the right
J+2k+ *** tir=h, ond where m=j+tk+4 " 4r-1,
T —— o S———— =

22 Smail, seo, 146,
ssronwioh, sec, 52, Smeil, sec, 177,