[HARKNEss] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 113 
§ 13. The series J7,+J7441+J%42+ ... is suggested as a 
generalization of the series in formula (24). 
The expression 
Gees (ee Re Gx ne: See À (31) 
For the left hand member 
ml) [1- S at s.s—1 F ] 
Ss Qvt+tstl  Q+s+1.2v+54+2 9° 
_ f2v+25 2s a | 
a ( s 2p+ 25 gS 5 =) 
2v+2s 
hence the J-series = aaa = aC": ) [:- ack SS 
S 2v +25 
Dom pram RES AG ) 
; > Le Gl) Caos) G ae 
Thus JF, + Sr + P,40 ooo SS fleas apy: +st1y? 
A similar method founded on the identity (32) 
6 ) nt Gs) WG) (a) nd 
= Gi ) (v+s) en} (33) 
where s ? 1, shows that 
2v 
yp J’, + (v1) Ji + (vp +2) Jo + = ols) 
Re (—1) Den-2) rte vts G \ 34 
ree 1) 2 iT o+s+1y? c= ee 
The presence of the factor 2y—1 accounts for ie simplicity of the 
formula (25), for when »=#, all terms after the first on the right hand 
side of (34) drop out. 
§ 14. Remembering that 1/T(x) is a transcendental integral func- 
tion, it foliows that the poles of 




I (vy) T (1—») fee Fe ST 1 
pty uv ~ sin pr u+v uv 
Ca) Bie) Gon) 
considered as a function of v are the points »=0, +1, +2, .... Also 
the poles are simple. Ho putting 
Ce ie 5 a5) So =e 
