THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 115 
CONTINUATION OF PaPER— 
“ THEOREMS RELATING TO A GENERALISATION OF THE BrssEL-FUNCTION.” 
(MS. received April 19, 1904.) 
8. 
The theorem 
Toy(2) Hn(6) - i sya) ae Ce 1p Mala) bu) ie eee 
Jae (a+ )(a + bp?) (a+ b)(a + bp?)(a + bp?)(a + bp*) _ wih: (a) 
[2 [2}*[4? es 
discussed in the first part of this paper may be obtained very naturally from the 
properties of a certain function analogous to the exponential function. Hlsewhere,* by 
means of the function E, I have obtained 
b a {fo Raye a 
J ofa =) — 2pJ| aila)a =) tals ernie +(—1)'2p*d, l@&a(=) =) TER see 
(a+b)?  @+0)(a + pio)? _ : -» (P) 
=l- P DP [4p oO 58 90-00 
We naturally expect to find some general form to which both (a) and (8) will belong, 
as particular cases. The following is the general theorem which will be obtained from 
the function E,, just as the addition theorem for Bessel coefficients is obtained by 
: : ‘ 1 
means of the exponential function. Exp. (5( = ;)) 
Jy (4, 6) = Jun (@)an(2p"™) - p pe eke ele” hag eek te Weal) a OP eee (y) 
J; (a,b) = 1- ee CES aa 
In case v= 0 we have the quasi-addition theorem (8). If, however, y=1 we have the 
quasi - addition theorem (a). The corresponding theorems for the function J* (a , b) 
will be briefly noticed. 
y _ (a+b)(a+ bp*) . . (a+ bp?) {, (a+ bp™”)(a+ bp”) 
Jn{, 8) TID. Baa] {! [2n+ 2] [2] 
mare Nate) _ (8) 
[2m +2] [2n+ 4] [2] [4] ww 
The expression for J’ (a,b) will be given also in the case when n is not a positive 
integer. 
* Proc. Lond, Math. Soc., shortly to be published. 
