GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. Ee 
2 
cannot be expressed as a product of n factors analogous to (A*x’—1)”, but it possesses 
the property 
de Nee KN eReey. [n]! 20-2 
- ce te Ga) = 
a, lene, * (Te =7}) @.a), 
06 2 — p3)(A? — p°) a ae (Cnt) : f : (25) 
(2), 
which is a particular case of 
* [P,*z]}"= 1+ ey ae ™ ya J : . (26) 
=i 
Peal 
substituting 
2S for z 
ON ay ae 
ald 
n a positive integer for m 
ees: S / — yrprrte L2] [20-2]... . [2n—2r+2)] 1 
eal 3 ee PI... 2 e 
which may be differently expressed 
(2 — p*)(d? — p°) f (2 — ptt) en N22 fon 
so. (2), — p*[n ae ye Q), Sod 0 8 GC ( ) 
If we use the symbol It” as the reverse of D”*” we have from (22) 
my (ot 2) = (A? — p8)(A2— p>)... . (A2— pt) ; 
I Pr (xP X) = ey (2), 2 (28) 
= pe — dN.» =v ean ee Vine ae r2 1)" 1 
Bs nee Speer p2 — 1 ie ee 
of which particular cases are 
I (n+) - ee (n+7) sy prt 5G yee [2n]! ¢ 
ta Maer ee Perm, «OO 
4. 
It has been shown in art. 9, Part I., that if 
ae seee dl] ers Sl) ae 
y= af a Mes le aToea 2m } (30) 
and f(x) denote 
—m—2v—3 [—n—v—3] p* [nm +yv ae 3] [n ar v+ 4| [—n—v—3] 
P [n+v+1] [nty+Q]A} ay = [203] e 5 ae ; 
then 
dy 1 dy 
da age * { He eA nv) fa + [n-v] [-2-v - Ly=S@) - Ka”) 
If we change to Ap” this differential equation is identical in form with the differential 
equation satisfied by P!"'(x,)). 
* Proc, Lond. Math. Soc., “ Series connected with the Enumeration of Partitions,” series 2, vol. i. 
TRANS. ROY. SOC. EDIN., VOL, XLI. PART I. (NO. 1). 3 
