[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 105 
The expression on the left-hand treated as a function of yw? has the 
sum of its residues equal to zero. 
yv+1 v+3 
a arse 16 SS peor 
oe y by v—1, this ar (14) 
+2 fe y+4 20 
v. re Seer a Ci =a .ytst+l1 ae oe eee PES EDR a 
The a eee not be overlooked that (14) can equally well be 
derived from (6) by equating the powers of p° on the two sides of the 
identity. 
Attention may also be called to the curious identity that arises 
when y is given the special value 0; viz. 
DES eel pee 1 
(v-+1)? (v+3)?...(v+2s+1)? v+1 v+l...r+s+l 
Ss 1 1 5 1 1 
—( See me (ane — .. (15) 
Returning to formula (14) we shall show that it can be readily 
converted into an identity connecting Gamma functions. When both 
sides are multiplied by 
4 x v+2s+1 
— ] —<——— = ; 
fee Ja T(v+4) s! G) 
the expression on the right hand side of (14) becomes 
(—1)s =! (v+1) T(v+1) 1 
Vets) LO 3.43 S!TO+s+2) 
= (v+3) T(v+2) 1 
NDS EE TIT GES ES) 
ie (v+5) T(v+3) 1 a x\? +2541 
218 .y+% s—2!T(v+5+4) iG) 
This is precisely the general term of 

ee asi Soe) Fy 
Vr T+) 5=0 S!S+5.v+5+3 CN CEA 
The expression on the left hand side of (14) Le 
v+2s+1 
CNE RARES Rae ESS — 
VE 2s Pt 2st Hee +)v+4 V+S+5 
a ore 
Nr (25-2) EE ey” 
eee Jar Q-28-1 1 ©) 
eee Se may GSH) 2 
(—1)s vt2s+1 
Sree à) T(vt+s+3 are ) 
1 : x y+2s4+1 
To 
= the general term of the series Z (x) = ZX T+) Po-rs+3) 


