Further Note on Factorizable Continuants. 



189 



7. The similar investigation of the case where the continuant is 

 of odd order leads to the following theorem : — 



The continuant of the (2m - l)th order 



7i A 2 /3 2 

 y 2 A 3 



is resolvable into linear factors, if, ■putting 



m - 1 7M w/t-1 



u*c /mve 



Ax = a + 7, A 2 # = a + ___ N, A 2W+ ! = a + - -— - N ; 



2(20-1)" 

 



n 

 y z = S-b, y 20 = T + -—B 1 , 



2m 



the factors being 



a + b + r 



a + b + t - - E 

 ni- 



2 

 a + 6 + r- --K 



m 



2(20 + 1) 



' + w(^ + l)(m-l ) 



72^+1 = -,^ ; - (0 + 1)E - - — b \; 

 m(26 + 1)\ m - 1 > 



7 1-n 



a + o — t -s + — B> 



m 



2 

 a+b—r-s+- E 



.777, 



3. 

 a + b-r -s+— R 



77£ 



7 m - 1 T ^ 

 a + 6 + t - E 



m 



(a + b-r-s+ m ~ 1 B, 



\ m 



(IV.) 



Here we have to note that 

 and hence that 



A x + A 2 + . . . + A 2m _! = (2m - 1) (a + 6) - (m - l)s + r, 

 -a result readily verified by looking at the sum of the factors. 



