GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 1l 
performing the operations indicated on the left side of the differential equation we 
obtain | 
{p[m,] [m -1] + {1 -[e-v]-[-2—v—1]} [mm] + [ev] [mv Tf Aga — Am, Jim, — LAr 
+similar expressions in Ms 
: Mz : : ; ‘ . (60) 
Since plm,|[m,- 1] = [m, }? —[m,] 
the expression which is the coefficient of A,x’’ in (60) reduces to 
{[74q] - [ev] {Lm ]-[e-v — TA, 
so that we have altogether the series 
{{om] —[>— v1 fom)— [— = — 1] } Aa — Ly] [rm = 1 or 
+ {[m,] —[n-v]}{[m.] -[-2—v—1}} Aga! — Am] fg - LA wP"™" 71. (61) 
Choose Ms = M,—2 
Also choose 
{[m,41] - [2 -v]}{[m41] -[-2-v -1)} A, =A [m,] [m,. - 1] A, 
Let m, also be so chosen that the coefficient of x"! may vanish, then 
m, =n—-v Or -n-v-1 
For the value Mm, =n-v 
Mrz, =n —v — 2 
we have 
Qy-2r-+1 eee) 25pes Dyn 1] 
a Dee [mn —v-2r+2][n-—v—-2r+ 2) 
Aris "dN, [2r] [Qn — 2741] Ce, 
and for the value m, =—-n-v-1 
Mpyy= —-n—-v—1—24r 
we have 
_ , B [etvt2r—1][n+v+2r] 
Arn =Ary2 [Br] [n+ 2r-+ 1] ae) 
From relation (62) we have the series 
praferrareater Ne Iarese nf 0 
a solution of the differential equation 
pa ga ty ay yd + +4 1-[n-v]-[-n-v-1] La + [nv] [-n-v-1y=sla) - fe") (65) 
dx dsc”) 
A(x) denoting the function 
[n—v][n-v-1]A { goa afl a ae =i Zi a Gg ha \ (66), 
