THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 113 
In terms of the function I’, of this paper, this theorem is 
1 Diy-e- 6D — 1 4 pf [6 | 
p* E((y=a)EAly-8) Teal a eae 
Change the base p to p* and put 
a= —m 
= —-m-n 
y=r—-m+l1 
we obtain 
TA([rt+m+n+1))T([r—m+1]) _ mar flee [2m] [2m + 2m] 94m on 4 
T.([7 + 1})Dya({[7 + 2 + 1)) Coe ae yn ee 
20T\ «1s hae Im—2s+2]-[2n+2n]..... 2m + In — 28+ 2) 90 osem+n 3 
+l ; [2] ay ie ee ae i ao 2m + 2s] » Sherer o> c> 
Now consider 
p> [2m ][Qm = 2][2m + 2n][2m + 2n — 2 
) 2m)\\ 2m + 2n 
J omen(A) — om [ en ae (A) + amein [2][4] Tier ON Tab ob (39), 
The coefficient of \”t”” is the infinite series 
(yee [ p? [2m] [2m + An] 
Qn + 2m + 2r} {Yr — Im}! : pimtn 19) (2r — 2m +2 BF tea a ] : . (40) 
2] 
which by (38) reduces to 
(aly Poll tn tr + 1/)Po((r - m+ 1)) (41) 
{2n + 2m + 2r}!{2Qr — 2m}! Dyo((7 + 1) o((7 +241) : ! 
Now remembering 
{2s}! = [2]}Tyo([s+1]) = (2).0,([s+ 1]) 
the expression (40) reduces to 
ja 2m(m+n) 
Trt In r+ IO Dass a a a 
which is (—1)-"p”""*™ x coefficient of \”*”" in the series J,,;. 
This establishes 
Mp 2m(m-+-n a 2 2 +2 % 
(= 1p (A) = Tamta(A) - ten | iu Sr BC tees. cad ink 5 (ee 
an extension of 
9 
i, Sea e ee a a a Cin ; ; . (44) 
Lome. defined J,, for negative integral values of 1, so as to make this theorem 
always hold: for example, suppose a negative integer, and put it equal to —m, then 
we have by this theorem 
Foy, . (45), 
extending 
Te Seis 
also 
dtm = (-1)"dog 
as may be shown by inverting’the base p in expression (43). 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 18 
