THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 117 
we obtain 
n=0 
bt x 0 Uta q v—1\4n oI Nyy (n-+v) Y v—1\f—n 
Ba (Pi (POE) = Spt bp) + (= 1p Op 
Bp [ ] BP [2] =O n=1 
+2 
= Dye Fm op)" - — (p) 
We have now, on taking the product of (A) and (), 
bt jou 
Seen Sartre = Bali Pol ~ tay )Pa(e)a( -"Tay) 
P 
The product of the four basic-exponential functions on the right of this expression is 
the product of two convergent series 
{ ‘ies (a+ b)t z (a+ )(a + pb)t? ce igre (a+ p*b)t eH (a+ p%b)\(a+ pee) tj (B) 
l x 
[2] (2) [4] pre Srieae [2] (2) [4] j 
This result follows from result (7) of article (9). 
If now we equate coeflicients of the various powers of ¢ in (B) with the corresponding 
coefhicients in 
+o +00 
D> n@)t" x pe a Oe ye 
remembering that 
Jim = (S 1)"J pin 
I= (-1)"I-ng 
we obtain from the terms which are independent of t 
Tio 2) Ipo(Op"*) — (B+ pT (a) Ip?) + (p?P”) + pO) Ta) Ip (bp?7) — os . (p) 
ACE le ee 
[2p 
which by an obvious reduction becomes 
‘ Me _| 4 : = sactenaesl AG : 9 
Tio(@)Ipo(OP ip: LS a) Iey(OD” “eaten ayy alte |y (a) Ipa(Bp" =) eaten . (c) 
[27] [27rv] 
=Ji (a, b) 
Hquating the coefficients of t” we obtain 
m=+2 
SS Sn) dr Ole or re a J; (a ) b) 0 ¥ . (tT) 
m=—co 
the expression for J, being that given in article (8) expression (0). 
11. 
When 7 is not a positive integer the expression 
(a+b)(atpb)..... (a+p”"~d) in J” (a, b) 
must be replaced by 
L @+4) at p'o)\a@+p%d) ... . (a+ pb) 
(=o(@e b(n pets) ea (ap tte) 
jp < Il 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 19 
