fY> tf] 
1902-3.] On the Series y— 1 + F([a] [/3] [ 7 ]) . p-p + , G ^ G - 441 
The following theorems will be required in subsequent work : 
j ct -j- h | 
1 n j 
= 
0 
-+r)i « 
| n- 
r}{ 
b {* 
r j 
00 
= 2^ 
0 
pr(a-n+ r ) | 
a (. 
n-r ) 
PIP 
-1]P“2]. 
W! 
. . . \b-r + 1] 
J . . (4) 
P> 1 
f a ) 
_ / a \ . 
p n - 
l P H 
- 1 - 1 . . . 
. jprc-r+1 _ J 
} n-r j 
1 n ) 
p&-n-\- 1 _ 
l * P a 
-»+ 2 - 1 . . . 
pd-n+r _ ^ 
This can be proved by substituting for j ^ | and j n ^ r j the 
infinite products which they represent. 
( 3 .) 
d 
If we operate on x with the operator we obtain 
~ !] 
d 
d{xP) 
[m] [m - 
d<r) 
■ xlml 
[m] [m - 1] \m - 2] . . . . [m - r + 1] xP r ^ m _r l 
= [m] [m - 1] [m - 2] [m - r + 1] a4 m J _ M 
Consider the series of operations denoted by 
ajM 
which is 
operate on this with 
we obtain 
and in general 
gives 
r = 00 / „ \ -v'l/’J 
operating on a£ m l we obtain 
[m] [m - 1] • • • • [m - r + 1] 
This series is convergent if p be greater than 1 and is x l m l | a m j- 
by theorem (4). 
* Proc. Lond. Math. Soc., vol. xxviii. p. 477, 
PEOC. EOY. SOC. EDIN. — VOL. XXIV. 
29 
