Page 232 - Determinants and Their Applications in Mathematical Physics

5.6 Distinct Matrices with Nondistinct Determinants 217

1 −x x 2
φ 1 (x + y) φ 2 (x + y)

= φ 0 (y) φ 1 (y) φ 2 (y) ,
φ 2 (x + y) φ 3 (x + y)
φ 1 (y) φ 2 (y) φ 3 (y)

1 −y y 2

= φ 0 (x) φ 1 (x) φ 2 (x) (5.6.19)

φ 1 (x) φ 2 (x) φ 3 (x)
1 −x x
2

φ 2 (x + y) φ 3 (x + y) 1 φ 0 (y) φ 1 (y) φ 2 (y)
=
φ 3 (x + y) φ 4 (x + y) −xφ 1 (y) φ 2 (y) φ 3 (y)
x φ 2 (y) φ 3 (y) φ 4 (y)
2
1 −2x 3x
2

1 −x x
2 3
.(5.6.20)
−x
φ 0 (y) φ 1 (y) φ 2 (y) φ 3 (y)
=
φ 1 (y) φ 2 (y) φ 3 (y) φ 4 (y)

Do these identities possess duals?
5.6.3 Determinants with Stirling Elements
Matrices s n (x) and S n (x) whose elements contain Stirling numbers of
the first and second kinds, s ij and S ij , respectively, are defined in
Appendix A.1.
Let the matrix obtained by rotating S n (x) through 90 ◦ in the

anticlockwise direction be denotes by S n (x). For example,
1
 
1
 10x 
1 6x 25x  .
 2 
S 5 (x)= 
1 3x 7x 15x
 2 3 
1 x x 2 x 3 x 4
Define another nth-order triangular matrix B n (x) as follows:
←
B n (x)=[b ij x i−j ], n ≥ 2, 1 ≤ i, j ≤ n,
where
j−1
1 j − 1 i−1
b ij = (−1) r (n − r − 1) , i ≥ j. (5.6.21)
(j − 1)! r
r=0
These numbers are integers and satisfy the recurrence relation
b ij = b i−1,j−1 +(n − j)b i−j,j ,
where
b 11 =1. (5.6.22)