FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 31 
The symbol { }! must not be dissociated from the expression Bin which it occurs, 
because {r}! as formed from n elements p, p, ... p, 18s not the same expression as 
{y\! formed from m elements p; Py - . - Pm- 
In general, for n elements 
{1}! = Pn 
{2}! = = (Dn + Pn— )Pn— 1 
ag (Pn + Pn— 1 + Pn—2 Prot = Dn=2 ea a = . . (12) 
At this point some properties of the coefficients 
(n)! 
(n—r)! {r}! 
re)! = 
>(- Gana 0 : , = (3) 
! 
S eae. 3 
( eae : 
As an example of this property take (p,, py, P3, Py) =(1,7,9, 5), then 
may be noted 
which is the generalisation of 
(4)! = (14-74+94+5)(7494+5)(9+5)5 = 22-21-14.5 
(3)! = (14. 7+9)(7+9)9 = 17-16-9 
(2)! = (14+7)7 = 87 
(1)! =1 
{1}! =5 
{2}! = (5+9)9 = 14.9 
{3} = G4947)9+47)7 = 2116-7 
; {4}! = (5+94741)(9+741)(741)-1 = 22-17-8-1 
The expression 
(ae (4)! (4)! 
; @Hoy ~ Bay * Ora” Mey * Oy 
is 
ee Ones SHEL 4) 735g 
OS 9 SO aaa a 
If p; po, . . . be an arithmetical sequence 1, 1+, 1+ 2a, etc., then 
(7)! _ (2+ (m—l)ay(2+na).. . . (2+ (2n—-2)a) 
agate {ryt ~ (l+a)(1+2a).... (1+(n-1)a) ; ie 
a generalisation of 
n! A 
Ds n—T! re 
to which the identity reduces when the sequence (p, pop3;---)=(1,1,1...). 
i theisequeneenisi(p, p>... )=(1,3,5, . . -) 
(n)! 2n—1 
2 eee ant =2 : ; Se (al) 
