[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 109 
(i) Let # be an even positive integer. 
We have 
P32 DY a oe 
= 242J)(2x) +2Jo(2x)P+...+2Jon(Qx) "+ .. 
= DS 2 (2%) t= 2-F 6 3 
2 Jon OL ee oes 
or 
A[Jo(x)+Se(x) P+... +S o(x) 1 2+#...12=2+27(2x) +... 
+ 27 on (2x) P+... 4927 (2x) 17 2+ 
Equate coefficients of ¢?” and we find formula (19) with # for y. 
(ii) Let x be an odd positive integer. The same process gives 
Jan (2%) = 4 [Fo Jan Je Jan-2+ . . + Jn-1Jn+1] +4 [Se Jonr2 
+ Ja Jenga oa | 
Combine this with 
à J n° (x) =4 [Jo Jon — Ji Jon-1+ Je Jon-2+ ... +In-1 Inti] 
+ 4[Si Jontit Je Jont2 + Ja Jonr3+ ...], 
and we find that 
Jon 20) = 2 JPA (Into In-2t Jura Inca... + Jon-1 Jon ti) 
A St Sensi t+JeJonvat. ) 
=2Iv+4 (Inte Tu-2+ Jura Jn-st+...ad inf.). 
§ 9. It is pe that 
1 pel 20 
(eae Gries FI. ces Hees FE aS os 
UE 3 Braga Tie ea 
Sr) Je ere FE DA Are Es RE ALU EE RNA 2 LU 
| 
+ - this becomes 

When both sides are multiplied by 
a5 
‘© 4307) OR alee pease (20) 
Schénholzer’s formula shows that 
72 S (— ae DS PDP 25+21+1 
ere) @) 
Hence 
x Eos a (Ge s ee ) ef BC aye. 
is the coefficient of x24+1 in + 3 (—1)° (25+1)J7,, (x). But by (20) 

Sec. III, Sig. 3 
