GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 13 
Parr II. 
iV; 
The series 
y=A ' gir — pil® a ciean 1] eas APeeoee oS \ 3 : (1) 
has a general term 
aes ee Penide el araeen hem fae Raeres ek [n-v—2r+1] 1 ie —v—2r 
[4] . . [2r]- [20-1] [2e—-3)..... [2n —2r +1] jar® 
A( = 1)"p" ne 
which may be written if n —v be positive and integral 
(- 1)"A 5 porte [n - v]! [nm]! ( 2) )n{ 20. - 2r]! 1 a aa 
Pa fe [7]! (2),(2),,_» pe 
so that if we give to the arbitrary constant A the value 
Nr [2m]! 
[nlm - v}@), 
we have the series 
T=@ Ian — 2r}! ee eae 
y= 2 7 1p a OS NS ae gS iC, 
which will be denoted by P*, (,\) 
When n and are not integral 
7 T((2n])"~ el eel 
C Supine. ° Bibi | 
When p=1 this function reduces to the function denoted w(nv) by TopHUNTER 
(Functions of Laplace, Lamé, and Bessel, p. 80). 
We see that 
TOP gheA) SUN Gren) é ' , (3) 
dal” re 
Also 
(v 1 
P_ (2A) =” ae =? ( pdao”r 4 ® ° . (4) 
= -- -\() 
dar’) | 
or 
Pp’ pb” ye Pinnl tA) 
ye) 0 ene 
For brevity P/., will often be used to denote Pr (ar). 
The function P*., satisfies the differential equation 
oR iy gigi. 
ae 7 ~ ep a + {1-[n-v]-[- n-v-1}e~ Pes fn ae NF 
y+2 v+2 p2 ) 
-5 at Pry(@d) ~ PLGA) 
ag ce ee ee 
Np” d(x?) da”) ; 
