24 THE REV. EF. H. JACKSON ON 
So the re-arrangement gives us an infinite series, of which the general term 1s 
qinter 1 
= oe 
2. ol 7! m+-Wrem+-Qr—-1l....... 2r+1 
which is the general term of S”J,(x) . ; : : - (69) 
S” indicating successive integrations. 
ci 
The preceding analysis shows us how to construct the analogous series for the 
generalised Bessel’s Functions. Consider the generalised form of VANDERMONDE’S 
theorem 
s=0 ml m—1l m—s--1l 
= >» s(z—m—sl) , | Eis lp =x coe Pp =) ; 70 
[a 1 Yn [2] nm ate a? p' — iT a p iss il “? pe a 1 Roe ( ) 
convergent for all values of m if p>1 : 
If m be a negative integer, 
pie oy dln Ba Bind a 
Pen [w+ ml][e+m—ll].... [x+d] ] : 
In the theorem (70) replace m by —m a negative integer, / by 2, x by 2r, and y by 2r. 
Then 
[27+ ee = (2ra oe Sere ae = e : eran oe x6: 
x 27] [Sve 'etilive eeene ces [27 — 2s + 2] (72) 
[2r+2m+2s]... . [27+2] 
Now [27+ 27] m is [47] =m 
1 
[47+ 2m] [4r+2m—-2].... [4742] 
and this may be written 
(2)or 
(2) or4m[27 + m0] - pares = 1 a fara 1] 
Dividing (72) throughout by [7]! [7]! (2), (2), we obtain 
1 1 ; Be. 
PEPE @)@), * [r+amy..... [47+2] 
1 S17) 8nsts-b2r [m +s—1]! (2)m—s—1 
Femi On 2.) | Pah swe. o... = 
the analogue of 
if Bac! — 2" 1 _m A 
27 .rlrlim+2r....7rtl m+rirl! arr Wamtrt ll r— 1! get 
ar! Fee, Si 
—— 
(74) 
used in art. 6 in the analysis of the series (65). 
At this point we define the function J,4,(aA) as the convergent infinite series 
on 
Fen gnh+2r_4 
= Atty g-1 d " f . : (75) 
pes [n+7]! [7]! (2),(2)n4r 
