100 
ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 
[119 
I ! 
It is of course easy, by the use of subscript letters and signs of summation, to 
present the preceding theorem under a more condensed form ; thus writing 
a x a 2 
.. a r 
1 Q‘ks + s- &k s -i Uks-i + s—l • 
•• a h~ a k,\ 
(¿2 
... a r . 
. a r+s j 
5+1 (. 
where k s , k s - 1} ...k 0 form a decreasing series (equality of successive terms not excluded) 
of numbers out of the system r, r — 1, ...3, 2, 1; the theorem may be written in the 
form 
a l a 2•• • a p—q+1 
x — a x x— a 2 ... x — a v = S o 
p Lcq a 2 a>r. 
x — at] x — a 2 ... x — a p _ q ; 
but I think that a more definite idea of the theorem is obtained through the notation 
first made use of. It is clear that the above theorem includes the binomial theorem 
for positive integers, the corresponding theorem for an ordinary factorial, and a variety 
of other theorems relating to combinations. 
Thus, for instance, if C q (a 1 ,...a p ) denote the combinations of a 1} ... a p , q and q 
together without repetitions, and H q (a li ...a p ) denote the combinations of a 1; ... a p , 
q and q together with repetitions, then making all the a’s vanish, 
and therefore 
x- a x ... x-a p = jS q o(~) q Cg(a 1} ... a p ) xP i; 
(x - ay = S q l(~) q C q (a,a... plures)(-)? a? xP~9, 
the ordinary binomial theorem for a positive and integral index p. 
So making all the a’s vanish, 
xP /S q oH q (œ x ... a p _ q+1 ) x «j x — a. 2 ... x &. p —q. 
If m be any integer less than p, the coefficient of x m on the right-hand side 
must vanish, that is, we must have identically 
0 = (~) q C' P —q—m ( a i> a 2) • • • &p—q) H q (3 1} CL. 2) ... g+i)- 
So also 
C P -m (ax, a 2 ,... a p ) = (-)? Op-g-™ (a x ,a 2 , ... a p _ q ) 1 p Q+1 
Suppose 
La, 
ai = 0, a 2 = 1 ... a p =p — 1 ; a x = k, a 2 = k— 1,... a p — k — p + 1, 
then 
tt 1} ... a p _ q+1 
k k — 1 . 
.. k — p + q 
a x a p 
q 
_0 1 
p- 1 -! 
W q 
q M 
-WAjW 10 “-■■■x-a p _q= [x]p~9; 
and hence 
[x+ky=s:M q [^[^-", 
v [ î]( 
the binomial theorem for factorials.