GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 5 
5. 
A very symmetrical differential equation is 
n_@ yon_@ Qn 
Uae Gem Tnle ) = ea Filed) ~ Tenlosdp) b re.) 
To prove this we must first show that 
 Trcn(@?Q) ~Trean(arn) = Pel s.(eap) (23) 
(n—1] (n+]] pra’ [ns . . . 
The coefficient of xt?"-™ in Jin _3(@?A) — J nga,(@?A) is 
Arter-1 Amter-1 
ar [7]! [m+ —1]! (2),(Q)ngra [7 — 1]! [+7]! (2), QDnge 
which is 
rte 
1 ue lees 
[7]! [a +7]! (2),(2)n4 eel +l) lle 1) \ 
eter pn] 
FE Teer}! 2),2) apd 
The r+1™ term of the resulting series is therefore 
[2%] NER pe aretha dl 
Me PERO 
which since 
gpln ter pel . gilt) 
x 
shows us that the series is 
» 
et Ton AP) ; (25) 
and we have established the relation (23) between three successive functions, reducing 
when p=1 to 
Nees ee =I, ; (26) 
Now it has been shown that 
hae, (aA) = = | DOS (aD) ; ‘ . - (27) 
Changing the independent variable « to x? " we obtain 
—n F d —N py, 
da” ONina Ger) = ae") { GP NY (Hd) ; ; : 5 (28) 
which is 
y es Ain d p— "in 
a aa "J, (ad) \ . (29) 
as can easily be verified independently. 
We have already shown that 
I d - 
J ins(#A) ws eae xP ("J (ZA) t : : : (30) 
