Determinants and Their Applications in Mathematical Physics

3.1 Cyclic Dislocations and Generalizations 17

Hence

n

∑

j=1

∣

∣C

1 C 2 ···C

∗

j

···Cn

∣

∣=

n

∑

i=1

n

∑

r=1

(1−δir)λir

n

∑

j=1

arjAij

=An

n

∑

i=1

n

∑

r=1

(1−δir)λirδir

=0

which completes the proof.

If

λ 1 n=1,

λir=

{

1 ,r=i− 1 ,i> 1

0 , otherwise.

that is,

[λir]n=

      

01

10 0

10 0

10 0

... ... ...

10

       n

,

thenC

∗

j

is the column vector obtained fromCjby dislocating or displacing

the elements one place downward in a cyclic manner, the last element in

Cjappearing as the first element inC

∗

j

, that is,

C

∗

j

=

[

anja 1 ja 2 j···an− 1 ,j

]T

.

In this particular case, Theorem 3.1 can be expressed in words as follows:

Theorem 3.1a.Given an arbitrary determinantAn, formnother deter-

minants by dislocating the elements in thejth column one place downward

in a cyclic manner, 1 ≤j≤n. Then, the sum of thendeterminants so

formed is zero.

If

λir=

{

i− 1 ,r=i− 1 ,i> 1

0 , otherwise,

then

a

∗

ij=(i−1)ai− 1 ,j,

C

∗

j=

[

0 a 1 j 2 a 2 j 3 a 3 j···(n−1)an− 1 ,j

]

T

.

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.