Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 141

The only element which remains in columnnis a 1 in position (n+1,n).

Hence,

En+1=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

h 11 h 12 ··· h 1 ,n− 1 vn 1 /n

h 21 h 22 ··· h 2 ,n− 1 vn 2 /(n+1)

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

(hn 1 −t)(hn 2 −t) ··· (hn,n− 1 −t) vnn/(2n−1)

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

(4.11.20)

It is seen from (4.11.3) (withi=n) that the sum of the elements in the

last column is unity and it is seen from the lemma that the sum of the

elements in columnjisx
2 j− 1
,1≤j≤n−1. Hence, after performing the

row operation

R

′

n=

n

∑

i=1

Ri, (4.11.21)

the result is

En+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

h 11 h 12 ··· h 1 ,n− 1 vn 1 /n

h 21 h 22 ··· h 2 ,n− 1 vn 2 /(n+1)

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

hn− 1 , 1 hn− 1 , 2 ··· hn− 1 ,n− 1 vn,n− 1 /(2n−2)

xx

3

··· x

2 n− 3

1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.11.22)

The final set of column operations is

C

′

j=Cj−x

2 j− 1

Cn, 1 ≤j≤n− 1 , (4.11.23)

which removes thex’s from the last row. The result can then be expressed

in the form

En+1=−

∣

∣h

(n)

∗

ij

∣

∣

n− 1

, (4.11.24)

where, referring to (4.11.5),

h

(n)

∗

ij

=h

(n)

ij

−

vnix

2 j− 1

n+i− 1

=vnix

2 j− 1

(

1

i+j− 1

−

1

i+n− 1

)

+δijt

=

(

vni

i+n− 1

)(

(n−j)x

2 j− 1

i+j− 1

)

+δijt

=−

(

vn− 1 ,i

n−i

)(

(n−j)x

2 j− 1

i+j− 1

)

+δijt

=−

(

n−j

n−i

)(

vn− 1 ,ix

2 j− 1

i+j− 1

−δijt

)

=−

(

n−j

n−i

)

̄h

(n−1)

ij

,

∣

∣h

(n)

∗

ij

∣

∣

n− 1

=−

∣

∣− ̄h

(n−1)

ij

∣

∣

n− 1

. (4.11.25)