Determinants and Their Applications in Mathematical Physics

228 5. Further Determinant Theory

identities in whichr=±1 form a dual pair in the sense that one can be

transformed into the other by interchangingHsandhs,s=0, 1 , 2 ,....

Examples.

(n, r)=(2,0):

∣

∣

∣

∣

H 2 H 3

H 1 H 2

∣

∣

∣

∣

=

∣

∣

∣

∣

h 2 h 3

h 1 h 2

∣

∣

∣

∣

;

(n, r)=(3,0):

∣ ∣ ∣ ∣ ∣ ∣

H 3 H 4 H 5

H 2 H 3 H 4

H 1 H 2 H 3

∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣

h 3 h 4 h 5

h 2 h 3 h 4

h 1 h 2 h 3

∣ ∣ ∣ ∣ ∣ ∣

;

(n, r)=(2,1):

∣

∣

∣

∣

H 3 H 4

H 2 H 3

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣

h 2 h 3 h 4

h 1 h 2 h 3

1 h 1 h 2

∣ ∣ ∣ ∣ ∣ ∣

;

(n, r)=(3,−1):

∣ ∣ ∣ ∣ ∣ ∣

H 2 H 3 H 4

H 1 H 2 H 3

1 H 1 H 2

∣ ∣ ∣ ∣ ∣ ∣

=

∣

∣

∣

∣

h 3 h 4

h 2 h 3

∣

∣

∣

∣

.

Conjecture 2.

∣

∣

∣

∣

Hn Hn+1

1 H 1

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

h 2 h 3 h 4 h 5 ··· hn hn+1

1 h 1 h 2 h 3 ··· hn− 2 hn− 1

1 h 1 h 2 ··· hn− 3 hn− 2

1 h 1 ··· hn− 4 hn− 3

··· ··· ··· ···

1 h 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

Note that, in the determinant on the right, there is a break in the sequence

of suffixes from the first row to the second.

The following set of identities suggest the existence of a more general

relation involving determinants in which the sequence of suffixes from one

row to the next or from one column to the next is broken.
∣
∣
∣
∣

H 1 H 3

1 H 2

∣

∣

∣

∣

=

∣

∣

∣

∣

h 1 h 3

1 h 2

∣

∣

∣

∣

,

∣

∣

∣

∣

H 2 H 4

H 1 H 3

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

∣

h 1 h 3 h 4

1 h 2 h 3

h 1 h 2

∣

∣

∣

∣

∣

∣

,

∣

∣

∣

∣

H 3 H 5

H 2 H 4

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

h 1 h 3 h 4 h 5

1 h 2 h 3 h 4

h 1 h 2 h 3

1 h 1 h 2

∣ ∣ ∣ ∣ ∣ ∣ ∣

,