Determinants and Their Applications in Mathematical Physics

5.6 Distinct Matrices with Nondistinct Determinants 217

∣

∣

∣

∣

φ 1 (x+y) φ 2 (x+y)

φ 2 (x+y) φ 3 (x+y)

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣

1 −xx

2

φ 0 (y) φ 1 (y) φ 2 (y)

φ 1 (y) φ 2 (y) φ 3 (y)

∣ ∣ ∣ ∣ ∣ ∣

,

=

∣ ∣ ∣ ∣ ∣ ∣

1 −yy

2

φ 0 (x) φ 1 (x) φ 2 (x)

φ 1 (x) φ 2 (x) φ 3 (x)

∣ ∣ ∣ ∣ ∣ ∣

(5.6.19)

∣

∣

∣

∣

φ 2 (x+y) φ 3 (x+y)

φ 3 (x+y) φ 4 (x+y)

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 −xx

2

1 φ 0 (y) φ 1 (y) φ 2 (y)

−xφ 1 (y) φ 2 (y) φ 3 (y)

x

2

φ 2 (y) φ 3 (y) φ 4 (y)

∣ ∣ ∣ ∣ ∣ ∣ ∣ =

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x 3 x

2

1 −xx

2

−x

3

φ 0 (y) φ 1 (y) φ 2 (y) φ 3 (y)

φ 1 (y) φ 2 (y) φ 3 (y) φ 4 (y)

∣ ∣ ∣ ∣ ∣ ∣ ∣

.(5.6.20)

Do these identities possess duals?

5.6.3 Determinants with Stirling Elements

Matrices sn(x) andSn(x) whose elements contain Stirling numbers of

the first and second kinds, sij and Sij, respectively, are defined in

Appendix A.1.

Let the matrix obtained by rotating Sn(x) through 90

◦

in the

anticlockwise direction be denotes by

Sn(x). For example,

S 5 (x)=











1

110 x

16 x 25 x

2

13 x 7 x

2

15 x

3

1 xx

2

x

3

x

4











.

Define anothernth-order triangular matrixBn(x) as follows:

Bn(x)=[

←→

bijx

i−j

],n≥ 2 , 1 ≤i, j≤n,

where

bij=

1

(j−1)!

j− 1

∑

r=0

(−1)

r

(

j− 1

r

)

(n−r−1)

i− 1

,i≥j. (5.6.21)

These numbers are integers and satisfy the recurrence relation

bij=bi− 1 ,j− 1 +(n−j)bi−j,j,

where

b 11 =1. (5.6.22)