254 
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 
[755 
that is, 
M 11 (a — /3) — M (a n — (3 n ) + a/3 (a. n 1 — /3 n *) = 0. 
But instead of a, /3, it is convenient to introduce the ratio A of the two roots, 
say we have a = A/3; we thence find 
giving 
(A, -t-1) ^ — ci -f- d, 
A/3 3 = ad — be, 
(A + l) 2 _ (a + df 
X ad —be 
for the determination of X, and then 
The equation thus becomes 
_a + d 
_ , 
a 
(a + d) X 
X + 1 
M n (X - 1) /3 - M (X n - 1) ¡3 n + (X n - X) /3 n+1 = 0, 
or we have 
M n = f n -^- {(x n -1) M - (x n — x) /3}. 
\ — 1 
It is convenient to multiply the numerator and denominator by X +1, viz. we 
thus have 
Mn = “ !) M + ( xn — X){M — (X+ 1) /3}]. 
The exterior factor is here 
1 fa + 
A 2 — 1 U + 1/ 
moreover (X + l)/3 is —a + d: hence 
M — { a, b ), 
I c, d I 
and 
M — (X + 1) /3 = ( a, b ) — ( a + d, 0 ), = ( — d, b ); 
c, d 
0 , a + d 
c , — a 
the formula thus is 
M n = 
1 (a + d\ n -' i(A, n+1 -l)( a, b )+(X n -X)( -d, b ) 
A 3 — 1 VA+1 
c, d 
c , — a 
viz. we have thus the values of the several terms of the wth matrix 
M n = ( a n , b n ); 
I Cn> d n |