GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 3 
Introducing a parameter \ we see that (6) and (7) may be stated more generally as 
ese R2qint21 D4gint4 
Ee Bre a te aaa 4 oe 
oa aypencea| | (2NpAy an Fa) [one] * (8) 
and 
party (@) 1 | 1- [n] - [- nj at es + [nr] [-n]y = NLP (ar) ft (9) 
In the series (8) give to the arbitrary constant A the value ua or generally 
2),1 70 |! 
(2),11([n)) I(r] then throughout the paper J,,,(x,) will denote the series 
rn” T=0 V2 gltt 27] 
(2),[7]! 2] (Ale 2h an f2) 20... [2n + 27] ay) 
which can be written in the form 
AMtHry[n+27] 
22.1] [n+7]! ; 7 : - (ly 
Since 
ee ae ay: ipa 
[2] [4] ... [27] = pa pe ei 
" ; Bet se pet ea ia 
pt+l- p+ PS IE er ane an aa fe 
= (2),r]! 
and in general (7 not integral) 
(p ts 1)'TL,.([7]) = (2),1,( [7]) 
3. 
It may be verified at once that 
d 
and in general that 
ey 
d(a?") \ tS y,(,d) i = Mingy(?-A) . - (13) 
for 
eS (@A) 
Arterginter)—l 
(2),(2)n+r17! [7 oF r|! 
Qrtergp”l 2r] 
ZOOL] '[n +7]! ey 
since 
pire x = mr oor = 
[n+ 2r]—[n] = re 2 = oreeer - = p"[2r] 
