Determinants and Their Applications in Mathematical Physics

4.8 Hankelians 1 105

It follows that

aji=aij,

so that Hankel determinants are symmetric, but it also follows that

ai+k,j−k=aij,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

aij=φi+j− 2. (4.8.3)

Then,

An=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

φ 0 φ 1 φ 2 ··· φn− 1

φ 1 φ 2 φ 3 ··· φn

φ 2 φ 3 φ 4 ··· φn+1

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

φn− 1 φn φn+1 ··· φ 2 n− 2

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

, (4.8.4)

which may be abbreviated to

An=|φm|n, 0 ≤m≤ 2 n− 2. (4.8.5)

In column vector notation,

An=

∣

∣

C 0 C 1 C 2 ···Cn− 1

∣

∣

n

,

where

Cj=

[

φjφj+1φj+2···φj+n− 1

]T

, 0 ≤j≤n− 1. (4.8.6)

The cofactors satisfyAji=Aij, butAij=F(i+j) in general, that is,

adjAis symmetric but not Hankelian except possibly in special cases.

The elementsφ 2 andφ 2 n− 4 each appear in three positions inAn. Hence,

the cofactor

∣ ∣ ∣ ∣ ∣ ∣ ∣

φ 2 ··· φn− 1

.

.

.

.

.

.

φn− 1 ··· φ 2 n− 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

(4.8.7)

also appears in three positions inAn, which yields the identities

A

(n)

12;n− 1 ,n

=A

(n)

1 n, 1 n

=A

(n)

n− 1 ,n;12

.

Similarly

A

(n)

123;n− 2 ,n− 1 ,n

=A

(n)

12 n;1,n,n− 1

=A

(n)

1 n,n−1;12n

=A

(n)

n− 2 ,n− 1 ,n;123

. (4.8.8)