362 
MATHEMATICS: D. N. LEHMER 
Written at length this is 
p 3 - Mp 2 + Np - 1 = 0 
where 
M = A k - 2 + B h -t + C k 
and 
N = Ak-2 Bk -i — Ak-i Bk-2~¥ Bk-iCk — BkCk-i Ak-2 Ck — A k Q_ 2 . 
We have proved the following theorem concerning this equation : 
Theorem II: The characteristic cubic of any periodic ternary continued frac- 
tion remains unaltered by any cyclic substitution of the partial quotients. 
The proof is again by induction. Using this theorem we may prove the fol- 
lowing recursion formulae: 
Theorem III: For all integer values of n>3k — 1 we have 
An = MA n - k - NA n -2k + AnSk, 
B n = MB n - k - NB n -2k + B n _ U , 
C n = MCn-k — NC n -2k + C„-2k- 
From Theorem III we see that the A's y 12'sand C'sare solutions of the fol- 
lowing linear difference equation with constant coefficients: 
u x -3 k — Mu x -2k +Nu x - k —u x =0. 
The theory of such equations is well understood (Boole's Finite Differences, 
p. 208). By referring to that theory we may write, 
A n = ZKixl (i = 1, 2, 3,.... 3k) 
where K 1} K 2 ,. . . .Kz are independent of n and x h x 2 , x n are the roots of 
x 3k -Mx 2k +Nx k - 1 = 0, 
and so are the kth roots of the roots of the characteristic cubic. 
This leads to the equation. 
A n = P n { k XP v u vn + pi^QrV™ + ffrR^T, v = 1,2,3,. . o>*=l, 
where P v , Q v , R v , are independent of n, and pi, p 2 . pz, are the roots of the 
characteristic cubic. Similar equations hold for B n and C n 
From this last result we obtain the remarkable theorem: 
Theorem IV: If the characteristic cubic has one root, pi, whose modulus is 
greater than the modulus of either of the other two roots then 
him (A n+k /A n ) = Lim (B n+k /B n ) = Lim (C n+k /C n ) = pi 
If the characteristic cubic has two imaginary roots whose common modulus is 
greater than the absolute value of the real root then the fractions A n+k /A n , 
B n +k/ B n , C n +k/ C n do not approach any limit as n increases beyond limit. 
