Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 135

Now, perform the row and column operations

R

′

i

=

i− 2

∑

r=0

(−1)

r

(

i− 2

r

)

Ri−r,i=n, n− 1 ,n− 2 ,..., 3 ,

C

′

j=

j− 1

∑

r=0

(−1)

r

(

j− 1

r

)

Cj−r,j=n− 1 ,n− 2 ,..., 2.

The result is

Y=(−1)

n

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∆φ 0 ∆

2

φ 0 ∆

3

φ 0 ··· ∆

n− 1

φ 0 1

∆

2

φ 0 ∆

3

φ 0 ∆

4

φ 0 ··· ∆

n

φ 0 ∆α 0

∆

3

φ 0 ∆

4

φ 0 ∆

5

φ 0 ··· ∆

n+1

φ 0 ∆

2

α 0

...................................................

∆

n

φ 0 ∆

n+1

φ 0 ∆

n+2

φ 0 ··· ∆

2 n− 2

φ 0 ∆

n− 1

α 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

,

where

∆

m

φ 0 =

φ

m+1

0

m+1

.

Transfer the last column to the first position, which introduces the sign

(−1)

n+1

, and then remove powers ofφ 0 from all rows and columns except

the first column, which becomes

[

1

∆α 0

φ 0

∆

2

α 0

φ

2

0

···

∆

n− 1

α 0

φ

n− 1

0

]T

.

The other (n−1) columns are identical with the corresponding columns of

the Hilbert determinantKn. Hence, expanding the determinant by elements

from the first column,

Y=−φ

n(n−1)

0

n

∑

i=1

[

K

(n)

i 1

∆

i− 1

α 0

]

φ

n−i

0

.

The proof is completed with the aid of (4.10.5) and (4.10.8) and the formula

for ∆

i− 1

α 0 in Appendix A.8. 

Further notes on the Yamazaki–Hori determinant appear in Section 5.8

on algebraic computing.

4.10.4 A Particular Case of the Yamazaki–Hori Determinant

Let

An=|φm|n, 0 ≤m≤ 2 n− 2 ,

where

φm=

x

2 m+2

− 1

m+1

. (4.10.29)