GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 9 
The coefficient of #”-*”*" in the expression 
5B d 1 
Ne al Prn(2A) oy ee we: ) 
A(x? ) | 
1s 
ee 2n — 27}! : oy [2m — Ir +1) [2m — 27° + 2] [27] [nm — 27+ 2] 
ss 1 r, mero \n 2r-+-2 [ ‘ = oP ak 27 ; \ 
(“le [7]! [n —7]! [x — 27]! (2),(2)n—r | lee [w—7r+1][n—2r+1][n—2r+2](p""H +1) 
Now since 
[n—r+1](p""t'4+1) = [20 - 274 2] 
the expression within the large bracket reduces to 
_op| 2n — 2r + 1] [27] 
eres 
Sate [n—27r+1] 
which is 
pn] [u+1] 
[n—2r+1] 
and we may write 
AP mn 2X) _ Pl? ) 
d(x? ) 
[2m — 2r]! 
=P US (-VP 8 ra eT 6) 
d2z2I 
Now consider 
CUP in(@?A) — €,Pin—1(@D) 
The coefficient of x”-*"*" in this expression is 
— 1 \tyrr +2) n—2r [2n = 2r]! —o,( — 1) ryt ber tyne th [2m — 27]! 
oS raat [7]! [m — r]! [nm — 27]! (2),(2),-» ose : [7 — 1]! [m —7]! [m -— 27 +1]! (2), (2)n_> 
which may be written 
Vt? 2-2 +2 [2n — 2r|! Gall C 1 [7] (po + 1) 
oe [r]! [nm -—r]! [2 — 27]! (2),(2),._» \° we Tyr [n= 2r+1] 
If now c be chosen as 
dn] 
and Gi 6 a 
Ap[n] 
the large bracket reduces to 
[7] [2 +1] 
[n —2r+1] 
and we have 
r=0 : [2n - 27]! git—2r-41] (5 1) 
MInlePaferd) —PrnFPnafer) = Zn Im+ Ue MO a fn — Br + TN Or 
and this series has been shown in (50) to be 
il 
FP (2A) APU PA) 
Na 
(a?) GE ) 
= Mn] | 2Ppg(vPA) — 2 Pun (any } ; ; (52) 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 1). 2 
