THEOREMS RELATING TO A GENERALIZATION OF BESSEL’S FUNCTION. 401 
Pies (-2)=1 : ‘ : : yh?) 
TEGO Te (aes) es AY slg po ee" GO | gn na) 
‘p ‘p [=p 1 —p*1—p4 
ay pal 
= Balt) ©) 
which reduce, when p=1, to exp (x) x exp (x)=1, 
aN AB al nied xt 
E, (*)) #, ( Sa) pet oe As Va é ; (5) 
= Loe 
= E,( =) . 6) 
2a (1 — @p-?”") : : : : spe ali) 
The product is absolutely convergent if |p| > 1. The series are convergent, however, 
if | p | > 1, and also for | p | < 1 provided wy 
It follows that 
B,(02)E,( ~ ot)E,(o%E,(— 0%)... B,(o"2)E,(— ota) —Eyn(amB) 8) 
al 
| ea) as 
The corresponding theorem in case p <1 is easily obtained by inversion of the base p. 
7 p-1 5 p-l LP (TE AlN Gare ll) eas 2 8 (p?"-1 Ae) 
=I {1 - xp | ; ' a) Oo) 
m=1 
E (22). x ( eae) = tr {1 a eh ; . | | 
eel 5 (oa) = | ee (11) 
Se 3) 
x ( eros ; 3 y eee) 
= 2 en 
=>" : 13 
cme 0 = Gel) ape) oe 
(Pe :) i Ba( 2 3) ~ : | 1 — 2p(x? + x?) + 2p*(at + w-4) — 2p%(a + 2~®) + | (14) 
Pp p P p (1 — p?”) sy eles ( Quen 0 
m=1 
On putting 2=e™, the series on the right becomes Jacosi's function 
L1G 
. . K’ 
in which pees 
[aC ia) eae anal) (15 
ae eae 
p=, ego y unrestricted 
(Cf. Proc. Edin. Math. Soc., vol. xxii. (8).) One l= 
E Y x unrestricted. 
