Page 218 - Determinants and Their Applications in Mathematical Physics

5.5 Determinants Associated with a Continued Fraction 203

where
1 b 1

−1 a 1 b 2

−1

a 2 b 3
. . .
K n = . . . . . . . (5.5.4)

−1 a n−3 b n−2

−1 a n−2 b n−1

−1 a n−1 +(b n /a n )

n
Return to H n+1 , remove the factor a n from the last column, and then
perform the column operation

C = C n + C n+1 .
n
The result is a determinant of order (n + 1) in which the only element in
the last row is 1 in the right-hand corner.
It then follows that
H n+1 = a n K n .
Similarly,
(n+1) (n−1)
H = a n K .
11 11
The theorem follows from (5.5.3).
Tridiagonal determinants of the form H n are called continuants. They are
also simple Hessenbergians which satisfy the three-term recurrence relation.
Expanding H n+1 by the two elements in the last row, it is found that

H n+1 = a n H n + b n H n−1 .
Similarly,
(n+1) (n) (n)
H = a n H + b n H . (5.5.5)
11 11 11
The theorem can therefore be reformulated as follows:
f n = Q n , (5.5.6)
P n
where P n and Q n each satisfy the recurrence relation
(5.5.7)
R n = a n R n−1 + b n R n−2
with the initial values P 0 =1, P 1 = a 1 + b 1 , Q 0 = 1, and Q 1 = a 1 .

5.5.2 Polynomials and Power Series
In the continued fraction f n defined in (5.5.1) in the previous section,
replace a r by 1 and replace b r by a r x. Then,
1 a 1 x a 2 x a n−1 x a n x
f n = ···
1+ 1+ 1+ 1+ 1