Ji'+ljJ+f i-^V.-o. (11 
BESSEL FUNCTIONS 
239 
2. The following linear relation exists between three consecutive 
Bessel functions: 
For 
Jn-t 
j n+1 O) = ~ l J n (x) - <4-10) n > 0. 
X 
•2s+71—1 
y\7l 1 
+ 2(-l) 
x 
2 n ~ l (n — 1)! t=i J 2 n+2s_1 s! (n — 1 + «)! 
j n+1== -^(-iy. 
s=1 
Hence 
«^-1 + '^n+l 
~2s+n—1 
r n—1 
■ + S(-1)< 
2 n+2s “i(s — 1)! (n + s)! 
L 
1-1 (s!( n— 1 
(5 
(6 
a 
v,2s+ra—1 
2 n_1 (w—1)! i v y 2 n+2s_1 \ s\n— 1 +s)! (s — l)!(w+s)!) 
n—1 00 _ ^¿s + n—1 
* - + n2(-l)* 
n 
2»+ 2 *-:i s ! (w + s) 
(8 
™2s+n 
— — 1 V — • 
xo K J 2" +2 - 1 s!(w + s)! 
2 n T s \ 
= ^«O). . 
X 
3. We show next that 
2/^)= ( 7 n _ 1 (2:)-J n+1 (a:) w>0. 
For subtracting 7) from 6) gives 
T —T xnl Vf ^ + 
n-i ' n+1 — 2 n_1 (w — 1)! i 2 n+2s ~ 1 s!(w + s)! 
2 n+2s-1 «!(w + s)! 
= 2 J' n . 
From 8) we get, on replacing e/„ +1 by its value as given by 5): 
^(aO=-V.(*)+ *-,(*), »>0. (9 
X 
From 5) we also get 
•iW=*i,W-4,W «>0- 0° 
X 
4. The Bessel function satisfies the following linear homo 
geneous differential equation of the 2° order: