Page 243 - Determinants and Their Applications in Mathematical Physics

228 5. Further Determinant Theory

identities in which r = ±1 form a dual pair in the sense that one can be
transformed into the other by interchanging H s and h s , s =0, 1, 2,... .
Examples.

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

H 2 h 2
H 3 = h 3 ;
H 1 H 2 h 1 h 2

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

H 3 H 4 h 3 h 4

H 5 h 5
;
H 2 H 3 = h 2 h 3

H 4 h 4
H 1 H 2 H 3 h 1 h 2 h 3

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

h 2 h 3

H 3
h 4
= h 1 h 2
H 4 ;
H 2 H 3
h 3
1
h 1 h 2
(n, r)=(3, −1):

H 2 H 3

H 4 h 3
H 1 H 2
= h 4 .
H 3 h 2 h 3
1 H 1 H 2
Conjecture 2.

h 2 h 3 h 4 h 5 ··· h n h n+1
1 h 1 h 2 h 3 ··· h n−2

h n−1

1 h 1 h 2 ··· h n−3
h n−2
H n
= .
H n+1
1 H 1 1 h 1 ··· h n−4 h n−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 1

H 3 = h 3 ,
1 H 2 1 h 2

h 1 h 3

H 2
h 4
= 1 ,
h 2
H 4
H 1 H 3
h 3
h 1 h 2

h 1 h 3 h 4 h 5

1

H 3 h 2 h 3 h 4

H 5 ,
=
H 2 H 4 h 1 h 2 h 3

1 h 1 h 2