Page 160 - Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 145

where
n−1

C = λ n−1,n−1 C n +

n λ n−1,j−1 C j
j=1
n−1

T
= λ n−1,j−1 ν j−1 ν j ··· ν n+j−3 ν n+j−2
n
j=1
=2 −(4n−5) 00 ··· 01 T . (4.11.42)
n
Hence,
A n =2 −4(n−1)+1 A n−1
A n−1 =2 −4(n−2)+1 A n−2 (4.11.43)
.......................................
A 2 =2 −4(1)+1 A 1 , (A 1 = ν 0 =1).
The theorem follows by equating the product of the left-hand sides to the
product of the right-hand sides.
It is now required to evaluate the cofactors of A n .
Theorem 4.49.
(n) −(n−1)(2n−3)
a. A =2 λ n−1,j−1 ,
nj
(n) −(n−1)(2n−3)
b. A =2 ,
n1
c. A nj =2 2(n−1) λ n−1,j−1 .
n
Proof. The n equations in (4.11.40) can be expressed in matrix form as
follows:
A n L n = C , (4.11.44)

n
where

T
L n = λ n0 λ n1 ··· λ n,n−2 λ n,n−1 . (4.11.45)
n
Hence,
L n = A −1 C
n n
= A −1 A (n) C
n ji n
n
=2 (n−1)(2n−1)−2(n−1) T , (4.11.46)
A n1 A n2 ··· A n,n−1 A nn
n
which yields part (a) of the theorem. Parts (b) and (c) then follow
easily.
Theorem 4.50.
n−1

(n) −n(2n−3) 2i−3
A =2 2 λ i−1,j−1 + λ r−1,i−1 λ r−1,j−1 , j ≤ i<n−1.
ij
r=i+1