Page 120 - Determinants and Their Applications in Mathematical Physics

4.8 Hankelians 1 105

It follows that
a ji = a ij ,

so that Hankel determinants are symmetric, but it also follows that
a i+k,j−k = a ij , k = ±1, ±2,... . (4.8.2)
In view of this additional property, Hankel determinants are described as
persymmetric. They may also be called Hankelians.
A single-suffix notation has an advantage over the usual double-suffix
notation in some applications.
Put

a ij = φ i+j−2 . (4.8.3)
Then,

φ 0 φ 1 φ 2 ··· φ n−1

φ 1 φ 2 φ 3 ··· φ n

A n = φ 2 φ 3 φ 4 ··· φ n+1 , (4.8.4)

.............................

φ n−1 φ n φ n+1 ··· φ 2n−2 n
which may be abbreviated to
A n = |φ m | n , 0 ≤ m ≤ 2n − 2. (4.8.5)
In column vector notation,

A n = C 0 C 1 C 2 ··· C n−1 ,

n
where

T
C j = φ j φ j+1 φ j+2 ··· φ j+n−1 , 0 ≤ j ≤ n − 1. (4.8.6)
The cofactors satisfy A ji = A ij , but A ij = F(i + j) in general, that is,
adj A is symmetric but not Hankelian except possibly in special cases.
The elements φ 2 and φ 2n−4 each appear in three positions in A n . Hence,
the cofactor

φ 2 φ n−1
···
. .

. . (4.8.7)

. .

φ n−1 ··· φ 2n−4
also appears in three positions in A n , which yields the identities
(n) (n) (n)
A = A = A n−1,n;12 .
12;n−1,n 1n,1n
Similarly
(n) (n) (n) (n)
A = A 12n;1,n,n−1 = A = A n−2,n−1,n;123 . (4.8.8)
123;n−2,n−1,n 1n,n−1;12n