Determinants and Their Applications in Mathematical Physics

5.1 Determinants Which Represent Particular Polynomials 177

2.Hn(x)=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 x 2

12 x 4

12 x 6

12 x 8

··· ··· ··· ···

2 x 2 n− 2

12 x

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

3.Pn(x)=

1

n!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

x 1

13 x 2

25 x 3

37 x 4

··· ··· ··· ···

(2n−3)xn− 1

n−1(2n−1)x

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

(Muir and Metzler).

4.Pn(x)=

1

2 nn!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2 nx 2 n

1 −x

2

(2n−2)x 4 n− 2

1 −x

2

(2n−4)x 6 n− 6

1 −x

2

(2n−6)x

··· ··· ···

4 xn

2

+n− 2

1 −x

2

2 x

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

5.Let

An=|aij|n=

∣

∣

C 1 C 2 C 3 ···Cn

∣

∣

,

where

aij=u

(j−1)

=







(

j− 1

i− 2

)

v

(j−i+1)

, 2 ≤i≤j+1,

0 , otherwise,

and let

C

∗

j=

[

O 2 a 2 ja 3 j···an− 1 ,j

]T

.

Prove that

C

′

j

+C

∗

j

=Cj+1,

A

′

n+A

(n+1)

n+1,n=0

and hence prove that

An=(−1)

n+1

v

n

D

n− 1

(

u

v

)

.