Page 156 - Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 141

The only element which remains in column n is a 1 in position (n +1,n).
Hence,

h 11 h 12 ··· h 1,n−1 v n1 /n

h 21 h 22 ··· h 2,n−1
.
v n2 /(n +1)
E n+1 = −
.....................................................
(h n1 − t)(h n2 − t) ··· (h n,n−1 − t) v nn /(2n − 1)

n
(4.11.20)
It is seen from (4.11.3) (with i = n) that the sum of the elements in the
last column is unity and it is seen from the lemma that the sum of the
elements in column j is x 2j−1 ,1 ≤ j ≤ n − 1. Hence, after performing the
row operation
n

R = R i , (4.11.21)

n
i=1
the result is

h 11 h 12 ··· h 1,n−1 v n1 /n

h 21 h 22 ··· h 2,n−1 v n2 /(n +1)
E n+1 = ................................................ . (4.11.22)

h n−1,1 h n−1,2 ··· h n−1,n−1 v n,n−1 /(2n − 2)
x x ··· x 1
3 2n−3
n
The final set of column operations is
C = C j − x 2j−1 C n , 1 ≤ j ≤ n − 1, (4.11.23)

j
which removes the x’s from the last row. The result can then be expressed
in the form
(n) ∗
E n+1 = − h , (4.11.24)

n−1
ij
where, referring to (4.11.5),
(n) ∗ (n) v ni x 2j−1
h = h −
n + i − 1
ij ij

1 1
2j−1
= v ni x − + δ ij t
i + j − 1 i + n − 1

2j−1
(n − j)x
= v ni + δ ij t
i + n − 1 i + j − 1

2j−1
(n − j)x
= − v n−1,i + δ ij t
n − i i + j − 1

2j−1
n − j v n−1,i x
= − − δ ij t
n − i i + j − 1

n − j (n−1)
h
= − ¯ ,
n − i ij
(n) ∗ ¯ (n−1)

h = − −h . (4.11.25)
n−1 n−1
ij ij