Determinants and Their Applications in Mathematical Physics

216 5. Further Determinant Theory

1 − 4 x 10 x

2

1 − 3 x 6 x

2

− 10 x

2

1 − 2 x 3 x

2

− 4 x

3

5 x

4

1 −xx

2

−x

3

x

4

−x

5

1 α 0 α 1 α 2 α 3 α 4 α 5

1 −xα 1 α 2 α 3 α 4 α 5 α 6

− 2 xx

2

α 2 α 3 α 4 α 5 α 6

3 x

2

−x

3

α 3 α 4 α 5 α 6

(5.6.13)

These determinants areB
66
s
,s=1, 2 ,3, as indicated at the corners of the

frames.B
66
1
is symmetric and is a bordered Hankelian. The dual identities

are found in the manner described in Theorem 5.16.

All the determinants described above are extracted from consecutive rows

and columns ofMorM

∗

. A few illustrations are sufficient to demonstrate

the existence of identities of a similar nature in which the determinants are

extracted from nonconsecutive rows and columns ofMorM

∗

.

In the first two examples, either the rows or the columns are nonconsec-

utive:

∣

∣

∣

∣

φ 0 φ 2

φ 1 φ 3

∣

∣

∣

∣

=−

∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x

α 0 α 1 α 2

α 1 α 2 α 3

∣ ∣ ∣ ∣ ∣ ∣

, (5.6.14)

∣

∣

∣

∣

φ 1 φ 3

φ 2 φ 4

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x

1 α 0 α 1 α 2

−xα 1 α 2 α 3

x

2

α 2 α 3 α 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 3 x

1 −xx

2

−x

3

α 0 α 1 α 2 α 3

α 1 α 2 α 3 α 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

.(5.6.15)

In the next example, both the rows and columns are nonconsecutive:

∣

∣

∣

∣

φ 0 φ 2

φ 2 φ 4

∣

∣

∣

∣

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x

α 0 α 1 α 2

1 α 1 α 2 α 3

− 2 xα 2 α 3 α 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

. (5.6.16)

The general form of these identities is not known and hence no theorem is

known which includes them all.

In view of the wealth of interrelations between the matricesMandM

∗

,

each can be described as the dual of the other.

Exercise.Verify these identities and their duals by elementary methods.

The above identities can be generalized by introducing a second variable

y. A few examples are sufficient to demonstrate their form.

φ 1 (x+y)=

∣

∣

∣

∣

1 −x

φ 0 (y) φ 1 (y)

∣

∣

∣

∣

=

∣

∣

∣

∣

1 −y

φ 0 (x) φ 1 (x)

∣

∣

∣

∣

, (5.6.17)

φ 1 (y)=

∣

∣

∣

∣

1 x

φ 0 (x+y) φ 1 (x+y)

∣

∣

∣

∣

, (5.6.18)