Determinants and Their Applications in Mathematical Physics

5.1 Determinants Which Represent Particular Polynomials 175

But

a

′

ij

+ai− 1 ,j=

[(

j− 1

i− 1

)

+

(

j− 1

i− 2

)]

u

(j−i+1)

−

[(

j− 1

i− 2

)

+

(

j− 1

i− 3

)]

v

(j−i+2)

=

(

j

i− 1

)

u

(j−i+1)

−

(

j

i− 2

)

v

(j−i+2)

=ai,j+1. (5.1.13)

Hence,

C

′

j+C

∗

j=Cj+1, (5.1.14)

A

′

n=

n

∑

j=1

∣

∣

C 1 C 2 ···C

′

jCj+1···Cn−^1 Cn

∣

∣

n

,

A

(n+1)

n+1,n

=−

∣

∣C

1 C 2 ···CjCj+1···Cn− 1 Cn+1

∣

∣

n

. (5.1.15)

Hence,

A

′

n+A

(n+1)

n+1,n

=

n

∑

j=1

∣

∣

C 1 C 2 ···(C

′

j−Cj+1)Cj+1···Cn

∣

∣

n

=−

n

∑

j=1

∣

∣C

1 C 2 ···C

∗

j

···Cn

∣

∣

=0

by Theorem 3.1 on cyclic dislocations and generalizations in Section 3.1,

which proves (a).

ExpandingAn+1by the two elements in its last row,

An+1=(u−nv

′

)An−vA

(n+1)

n+1,n

=(u−nv

′

)An+vA

′

n

=v

[

A

′

n

+

(

u

v

−

nv

′

v

)

An

]

,

yAn+1

v

n+1

=

y

v

n

[

A

′

n

+

(

y

′

y

−

nv

′

v

)

An

]

=

yA

′

n

v

n

+

(

y

v

n

)′

An

=D

(

yAn

v

n

)

=D

2

(

yAn− 1

v

n− 1

)

=D

r

(

yAn−r+1

v

n−r+1

)

, 0 ≤r≤n