12 THE REY. F. H. JACKSON ON 
We also have from relation (63) 
voy) | peereri [mtv +1] [n+v+2] ee 
yoAle +p [2] [22 +3] Na z ea one } . “6F 
a solution of the differential equation in which /(«) denotes 
ervey! f-n-v-8) 4 pal @ tv +3] (m+vt4], 21 n—v=3) 4 ; 68 
an. e + pr [2] [20 +3] “a ee (68) 
Since \ is quite arbitrary, replace it by \p" and we then have the series 
n—v\|n— 1 e a 
y=eonst. | AP vg it) [ abs =i J payne 2gin-v ats ee i j (69) 
a solution of 
sally dl MY eet 
ra SD + { 1-(n-v]-[-x-v-1] peel (x —v][-—n—v—l]y=f(2) - f(a”) (70) 
The series (69) is when » is integral, and <n 
dP, i 2% (71) 
day 
The series and differential equation are analogous to one given by Heine 
(l - 24 - Xv + 1). OY + (wey DG=ny=0 (72) 
of which the primitive is 
Fey ease es =n . _ (78) 
ae ae 
The sum of the series of coefticients in Heine’s series 
4 (w= v)(m=v—1) (x —v)(n—v —1)(n—v—2)(n—v—3) _ 
‘g 2-2 —1 Pe a a ay rn ns 
is shown in Chrystal’s Algebra, Part Ll. page 185, or 209 (2nd edition), to be 
gn a+! 
~ yt Qn! ‘ : ; es) 
The analogous theorem in the general series is 
eeees ere weal —v|{~-v-—1][n-v-2][n-v-3 
Sais [2] [22-1] gee - [4] ae 1} [2 aes ie 
Qn (n]i fue 
(2),° [v]! [Qn]! : : . » (75) 
