Determinants and Their Applications in Mathematical Physics

5.8 Some Applications of Algebraic Computing 229

∣ ∣ ∣ ∣ ∣ ∣

H 2 H 4 H 5

H 1 H 3 H 4

1 H 2 H 3

∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣

h 2 h 4 h 5

h 1 h 3 h 4

1 h 2 h 3

∣ ∣ ∣ ∣ ∣ ∣

,

∣

∣

∣

∣

H 3 H 5

H 1 H 3

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

h 1 h 3 h 4 h 5

1 h 2 h 3 h 4

h 1 h 2 h 3

1 h 1

∣ ∣ ∣ ∣ ∣ ∣ ∣

. (5.8.2)

5.8.3 Hankel Determinants with Hankel Elements....

Let

An=|φr+m|n, 0 ≤m≤ 2 n− 2 , (5.8.3)

which is an Hankelian (or a Turanian).

Let

Br=A 2

=

∣

∣

∣

∣

φr φr+1

φr+1 φr+2

∣

∣

∣

∣

. (5.8.4)

ThenBr,Br+1, andBr+2are each Hankelians of order 2 and are each

minors ofA 3 :

Br=A

(3)

33 ,

Br+1=A

(3)

31

=A

(3)

13

,

Br+2=A

(3)

11. (5.8.5)

Hence, applying the Jacobi identity (Section 3.6),

∣

∣

∣

∣

Br+2 Br+1

Br+1 Br

∣

∣

∣

∣

=

∣

∣

∣

∣

A

(3)

11 A

(3)

13

A

(3)

31 A

(3)

33

∣

∣

∣

∣

=A 3 A

(3)

13 , 13

=φ 2 A 3. (5.8.6)

Now redefineBr. LetBr=A 3. Then,Br,Br+1,...,Br+4are each second

minors ofA 5 :

Br=A

(5)

45 , 45

,

Br+1=−A

(5)

15 , 45

=−A

(5)

45 , 15

,

Br+2=A

(5)

12 , 45 =A

(5)

15 , 15 =A

(5)

45 , 12 ,

Br+3=−A

(5)

12 , 15

=−A

(5)

15 , 12

,

Br+4=A

(5)

12 , 12

. (5.8.7)