Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 147

Proof. By interchanging first rows and then columns in a suitable

manner it is easy to show that

En=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

ν 0 ν 1 ν 2 ···

ν 1 ν 2 ν 3 ···

ν 2 ν 3 ν 4 ···

ν 1 ν 2 ···

ν 2 ν 3 ···

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.11.53)

Hence, referring to Theorems 4.11.5 and 4.11.6b,

E 2 n=(−1)

n

AnA

(n+1)

n+1, 1

=(−1)

n

2

−(2n−1)

2

,

E 2 n+1=(−1)

n

An+1A

(n+1)

n+1, 1

=(−1)

n

2

− 4 n

2

. (4.11.54)

These two results can be combined into one as shown in the theorem which

is applied in Section 4.12.1 to evaluate|Pm(x)|n.

Exercise.If

Bn=

∣

∣

∣

∣

(

2 m

m

)∣

∣

∣

∣

n

, 0 ≤m≤ 2 n− 2 ,

prove that

Bn=2

n− 1

,

B

(n)

ij

=2

2[n(n−1)−(i+j−2)]

A

(n)

ij

,

B

(n)

n 1

=2

n− 1

4.11.4 A Nonlinear Differential Equation..........

Let

Gn(x, h, k)=|gij|n,

where

gij=

{

x

h+i+k− 1

h+i+k− 1

,j=k

1

h+i+j− 1

,j=k.

(4.11.55)

Every column inGnexcept columnkis identical with the corresponding

column in the generalized Hilbert determinantKn(h). Also, let

Gn(x, h)=

n

∑

k=1

Gn(x, h, k). (4.11.56)