Page 32 - Determinants and Their Applications in Mathematical Physics

3.1 Cyclic Dislocations and Generalizations 17

Hence
n n n n

∗

C 1 C 2 ··· C ··· C n =
j (1 − δ ir )λ ir a rj A ij
j=1 i=1 r=1 j=1
n n

= A n
i=1 r=1 (1 − δ ir )λ ir δ ir
=0
which completes the proof.
If
λ 1n =1,

1,r = i − 1, i > 1
λ ir =
0, otherwise.
that is,
0 1
 
 10 0 
10
 
 ,
 0 
1 0
[λ ir ] n = 
 0 
... ... ...
 
1 0
n
then C is the column vector obtained from C j by dislocating or displacing
∗
j
the elements one place downward in a cyclic manner, the last element in
∗
C j appearing as the first element in C , that is,
j
∗ T
.
C = a nj a 1j a 2j ··· a n−1,j
j
In this particular case, Theorem 3.1 can be expressed in words as follows:
Theorem 3.1a. Given an arbitrary determinant A n , form n other deter-
minants by dislocating the elements in the jth column one place downward
in a cyclic manner, 1 ≤ j ≤ n. Then, the sum of the n determinants so
formed is zero.
If

i − 1,r = i − 1, i > 1
λ ir =
0, otherwise,
then
∗
a =(i − 1)a i−1,j ,
ij

.
∗ T
C = 0 a 1j 2a 2j 3a 3j ··· (n − 1)a n−1,j
j
This particular case is applied in Section 4.9.2 on the derivatives of a
Turanian with Appell elements and another particular case is applied in
Section 5.1.3 on expressing orthogonal polynomials as determinants.