THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION,. 109 
and obtain, after obvious reductions, 
= = p= Se 1 ) 
et erence = (1+p") | 1 feel (De 6 als 1)(p?”-? — 1) pt 
Cen t=?) 3 (1 =p") p-l 2 (p?-1)(p?-1) 2 
eee tt 
EN) Vp 1) 2 
In the same way, if we put 
a=p 
Cp 
Dea FHee 
iS Sy 
we obtain, after reductions 
Zaft Ss 
S,=1 aoa ser ete outer 
_ _ Cee) ane Cea) {1- el ee pee Ba. (19) 
ee =e. - (-p") | @-h@ +h? *@-DeP-De+ Te? a ae 
So that S,+ P- De — S, may be written 
(Glee 2) sez eee aa) 2 Seminal en (Bee PY ee oa sk 
Ceres lp)... (’=p™) jit (p?—1)(p? = 1) 
- =o (p= 1)(p? = 1) ns ; } 20 
Pap + ay } (20) 
Adding the terms with like numerators together, we obtain 
(l+p)(1+p?)..... (1+p") {o-2 ype 
ee Comes. ; - -eD) 
Ce ae ae) oes (1 - p?’) 2 —1 
ea, (a Se 
The series within the large brackets is the simplest type of series, and its sum is well 
known to be 
7 oS io) ere cea (l-p”) . ; 2) 
pale eae eee alee Al sadey 
"HPs Peeps 7 
_ 21 +p) +p%) ... 4p): (=p) 1-p) 2. (=p) 23) 
Ge UM) em 5 (ee) 
We have therefore 
Changing the base p to p® we obtain the series whose sum was sought 
1A), 8) [2] 2r— 2) 
[2] Br+2]  ? (4) Pre2}rta 
alee en) (er) ap )\—p)...d-") +» . ey 
(TE (Sv ee eee (1 —p* 7) 
The coefficient of \”" is obtained by multiplying this sum by 
{47}! 
{Qr}! {Qr}! {Ir}! {Ir}! 
