304 
ON A FUNCTIONAL EQUATION. 
[686 
where 
where 
viz. 
Gx n _ l + D — 
C (X n ~ l P + Q) + l) (\ n -'R + S) R'\ n + S' 
X n -'R + S 
RX 71 - 1 + S ’ 
CP DR 
) — C ( a + — ) + D [ c + —-), 
S' = CQ + DS, = C(-a-bX) + D (- c-dX); 
R'= W + trZ, S' = -W-XZ, 
A, 
where Z and W denote aC + cD and b(7 + dD as before. 
We hence obtain 
(j^n— 1 *^n) fit'll — 
-(AD-BC) 
C 
(X — 1) (A 2 — 1) (a + d) {ex 2 + (d — a) x — 6} 
X n 
(RX n + S) (R'X n + S') 
(AD - BC) 
C 
(X — 1) (A 2 — 1) (a + d) {ex 2 + (d — a) x — 6} (RS' — R'S) X 11 
X (RS'-R'S) (RX n + S)(R'X n + Sy 
or, substituting for RS' — R'S its value in the denominator, this is 
AD-BC (ad — be) (A 2 — 1) (RS'-R'S) X n 
(x n —i x n ) f3x n — 
C (a+ d) X (cD - dC) (RX n + S) (R'X 11 + S') 
*J{(a - d) 2 + 46c} (AD - BC) (RS' - R'S) X n 
and thence 
C (cD — dC) 
(RX n + S) (R'X n + S') ’ 
, _A V{(« — d) 2 + 46c} (AD — BC) v (RS'-R'S) X n 
<px ~ c x C (cD - dC) z (RX n + S) (R'X n + S') ’ 
the summation extending from 1 to oo. 
Now the before-mentioned integral formula gives 
1 _ 1 f sin (n log A + log k) t dt 
1 + kX n 2 J sinh 7rt 
1 _ f sin (n log A + log k') t dt 
1 + k'X n 2 J sinh 7rt 
R R' 
Taking the difference, and then writing k = , k' = , we have under the integral 
sign 
( Pv / jR / \ 
n log A + log ^ J t — sin \ n log A + log t,