Determinants and Their Applications in Mathematical Physics

142 4. Particular Determinants

Theorem 4.45 now follows from (4.11.24).

Theorem 4.46.

Gn=(−1)

n− 1

vnnKnHnHn− 1 ,

whereGnis defined in (4.11.10).

Proof. Perform the row operation

R

′

i=

n

∑

k=1

Rk

onHnand refer to the lemma. Rowibecomes

[

(x+t),(x

3

+t),(x

5

+t),...,(x

2 n− 1

+t)

]

.

Hence,

Hn=

n

∑

j=1

(x

2 j− 1

+t)H

(n)

ij ,^1 ≤i≤n. (4.11.26)

It follows from the corollary to Theorem 4.44 that

B

(n)

ij

=B

(n)

ji

=

n

∑

r=1

n

∑

s=1

H

(n)

jr

H

(n)

rs

K

(n)

si

. (4.11.27)

Hence, applying (4.11.7),

B

(n)

ij

=Knvni

n

∑

r=1

n

∑

s=1

vnsH

(n)

rsH

(n)

jr

i+s− 1

. (4.11.28)

Puti=n, substitute the result into (4.11.10), and apply (4.11.16) and

(4.11.24):

Gn=Knvnn

n

∑

r=1

n

∑

s=1

vnsH

(n)

rs

n+s− 1

n

∑

j=1

(x

2 j− 1

+t)H

(n)

jr

=KnvnnHn

n

∑

r=1

n

∑

s=1

vnsH

(n)

rs

n+s− 1

=−KnvnnHnEn+1. (4.11.29)

The theorem follows from Theorem 4.45.

4.11.3 Some Determinants with Binomial and Factorial

Elements

Theorem 4.47.

a.

∣

∣

∣

∣

(

n+j− 2

n−i

)∣

∣

∣

∣

n

=(−1)

n(n−1)/ 2

,