10 THE REV. F. H. JACKSON ON 
We see that if \ = p* 
and x = 1 
the left side of the above expression becomes identically zero, Therefore when 
A=p' and e=1 
[nr] { #P,,(@?r) — EP i.n() t 
must also be zero, which is 
Pry(l - p) = ptP_n(1 : p») : . : (53) 
from which we have for integral values of n 
Pal - pt) = p'Pio(1 + p}) 
Pp(1 - pt) = p!Pp(1 - pt) 
. . . . . (54) 
Ppl +p?) = ptPp—1(1 - p?) 
and so taking the product of the two sides of the equations 
Pon(1 pt) = p*Po(1 « pt) = p? : pe (595) 
which is 
[an]! p? patella Ul. [2 — 1] of] [2 -— 1] [n- 2] [n- 3] _ ees 
[7]! [7]! on ve Pla] [m—1] [m—1] ae EV ec2 eS ae we ey) 
; n||n a]! (2)n 
nants eer ‘Geen See * Lane mei aed Pee om 
the general term of the series being 
prt [m][m-1][n-2].... [n-2r +41] 
[2](4]. . [2r) . [Qn-1]... [22-2741] 
If we put p=1 this reduces to a well-known series 
al n-n-1l-n-2n-38 _ n!n! 2” 
2-2n-1 ° 2-4.9m-1-2n-3 "°° Om! 
10. 
The series 
n-v hips = =va1 n—v—2 
y= { gen al a ee a cae \ . (58) 
is a solution of a differential equation of the form 
wa) 5a?) 
porn at | (9 =v) = ea t ob + [ny] [—n-v-l]y=f(2) —f(a") (59) 
for, assuming that y can be expressed as a convergent or finite series of the form 
y = DyAdn 
