GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 
Then throughout subsequent work the function P,,,(a,) will denote 
ee [2n — 27]! 
2- Vana PO. ee (38) 
P.,-9(@A) denotes the series 
cei Qn — 2r']! 
=i ) LACS Ue see n—W—2,,[n—2r—2] 
2 ) [m — 7]! [m — 2x — 2]! [7]! @),.(2), “ oe » (39) 
We proceed to show that 
d-Pui(tr) dP (ar rd : 
Ee 2 nee Me Slant Pay). . (40) 
In Prrsr(@A) +«Pin-1(#A) the terms involving «”-”*™ oive us 
: 2n — 2r + 2]! wig = 
r.r+2 -1)’ [ Nn or, (2+1—2r] 
7 paren SN ONe es 
3 2 2n — 27}! a 
r—1.7-+1 Saye 1 [ Art1-2r, {2 —2r-+-1] 
es aa ars Ue 1c . 
which gives us an expression for the coefficient of #”-""™, viz., 
= Payt-T +2 [2n ae 2r]! | [2m = ar + 1] a 1-r [27] } eer tl 
CoMlg m—7|![m—2Qr]! [7]! (2),(2),_, | [wv -2r4+1 ca n—2r+1 
(2) 
The part within the large bracket is 
Dane = 1 ee Kp (p" = 1) 
(p- 1)[u- 274+1] 
Putting «= —»p this reduces to 
| p*[2n + 1] 
[w — 27+ 1] 
Therefore 
— : Qn — Ir}! p-*"[2n+ 1] Wate hs Pe 
= nV = — 1 yr? [2 n 27)! Pp [2 a+ rn 2r+1,[n—2r-+1] 
Pingr(@A) PPin-1(% p) pan YP [x = ra [n — — 27]! [r]! (2),(2),_ In Sorat 1} x 
=n i 1] [2n — 27]! n—2r-H1pln—2r+1 
1 rr+2 ON +11 +1) 
=e Oe. 
from which we a at once 
eee bps cg, rip lah, eG 
Aes = se) rae a = pe a+ ip a(® Pp) o : ( ) 
which leads to 
(41) 
peeks =p | [22 -—1]P,,_,(v?,Ap) + p*[Qn - 5]P,,_.(a?-Ap) + p'[2n — 9]P,_-5(w?-Ap) +--+: t (43) 
8. 
The relation between three consecutive functions is 
[7 ]Pr(wA) = AQ —1]aPp_y(w?A) — pp" [2-1] Piro (wr) . (44) 
which corresponds to the ordinary relation 
mP, = (2n—1)xP,_,-(n—1)Pn-s 
Consider 
AQn — VP a (ad) — «pln — V]Pp_n(@>) 
