GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 
The general term 
ae Sede PVE We ce te a ah gk pytr-i _ il 
Je Se earl ne a Bere tee... 5 BAP hz 4 
becomes 
tee ae eater PT peter] 
1B eS hae ms p27 — J] jes eaa see ae eee 2 pera 
ee ee Reeeeiiee [n+v + 27] 
Pl [4s 22 eiae+ay.-. . [2n + 27+ 1] 
and the infinite product 
(Bekas cal 
[P*2]" 
becomes 
seme al ha. oy ae a 
x=a| 2v|[2v — 2] [2-4] . . [2v—2«+2]-[-2n-3]... [-—2n-2«-1] 
so that we have this product 
{rz+vt+1][mt+v+2] % 
[2] [22+ 3] (2) [4] [2n 4+ 3] [204 5] 
T([2n + 1]) HQ (1-p) 
~ Ti({n +v))1({n}) - (2) 
Now take the differential hae 
i a Xp” dae se S| n-v—1]}o 2 + [n- v]{-n-v-1]y 
= f(a) — F(a”) 
pitty) [nt+v+2][n+v+3][n+v+4] 
=,1)7 
(36) 
(37) 
(38) 
(39) 
19 
and find a solution in the form of a series proceeding according to ascending powers 
of x. 
aie sw ‘sie: ie Ve 
Assume i Ayal] + Ayala) ak 
(40) 
Then performing the operations indicated on the left side of the differential equation 
(39) we obtain from a term Av”! the expression 
pl] [m — 1] Aan 
which is 
- xp" [m — 1]AaPt—" 4 11 —[n-v] -[-n-v—1]}[m] Aare 
+ [n-y] [n—-v- 1] Aa 
(41) 
{p[m][m—-1] + [m]-[m] [n-v] -[m] [-n-v-1]4+[n-v][-n-v- L]} Aa = pl” [ma — 1] Aan 
which reduces further to 
{{m] - [nv] }{[m] -[ =n —v— 1] aa 
= Agee 
(42) 
