1902—3.] On the Series y = 1 + F([a] [/3] [y]) . p-j + , etc. 439 
On the Series y = 1 + F([a][/i] [y]) • ppr, + F([a] [£] [y]) • 
x m 
F([ a ][/^][y+ l])r 9 jj + • • • * and its Differential Equa- 
tion. By the Rev. F. H. Jackson. 
(Read June 1, 1903). 
(i.) 
F([a] [/?] [y ]) denotes the convergent infinite product 
T (py~ a - 1) (py 
-a+1 
■a- P + 1 
1 ) (pt-a+K- 1 _ 1 ) . (py-P- 1 ) (py-fi+1- 1 ) . . 
- 1) . . (py-*- P +«~ 1 - 1) . (py - 1) ( py + 1 - 1 ) . . . 
. . . (py~P+ K ~ 1 - 1) 
. . . (^+*-1-1) * 1 
which may be expanded in an infinite series 
p > 1 
1 + py-*-P 
p a - 1 ’ pP—1 
p -1 - py- l 
+ p2(y-a-/3) 
p°- - 1 • jp ct + 1 — 1 ‘ 790 — 1 * P&+ 1 - 1 
p - 1 • p 2 - 1 ■ py - l ■ py +1 - 1 
p > 1 
When p = 1 the series reduces to the particular hypergeometric 
series F(a/3yl), and the infinite product reduces to 
r(y-q — ff)r(y) _ 
T(y-a)T(y- {3) 
The object of this paper is to obtain the differential equation 
which has for a solution the series 
2/ = l + F([a][/3][y])-g 
+ F(H M W) • F(W M [r + 1]) • jg + . . (1) 
X < 1 
P > 1 
orifp=l y-a-/3>0 
Proc. Lond . Math. Soc., vol. xxviii. p. 475. 
