Determinants and Their Applications in Mathematical Physics

4.9 Hankelians 2 121

Proof. The sum formula forTcan be expressed in the form

n

∑

j=1

ψr+i+j− 1 T

(n,r)

ij

=−δinT

(n+1,r)

n+1,n

, (4.9.10)

C

′

j=

[

(r+j)ψr+j(r+j+1)ψr+j+1···(r+j+n−1)ψr+j+n− 1

]T

n

.(4.9.11)

Let

C

∗

j=C

′

j−(r+j)Cj+1

=

[

0 ψr+j+1 2 ψr+j+2···(n−1)ψr+j+n− 1

]T

n

. (4.9.12)

Differentiating the columns ofT,

T

′

=

n

∑

j=1

Uj,

where

Uj=

∣

∣C

1 C 2 ···C

′

j

Cj+1···Cn

∣

∣

n

, 1 ≤j≤n.

Let

Vj=

∣

∣C

1 C 2 ···C

∗

j

Cj+1···Cn

∣

∣

n

, 1 ≤j≤n

=

n

∑

i=2

(i−1)ψr+i+j− 1 Tij. (4.9.13)

Then, performing an elementary column operation onUj,

Uj=Vj, 1 ≤j≤n− 1

Un=

∣

∣C

1 C 2 ···Cn− 1 C

′

n

∣

∣

=

∣

∣C

1 C 2 ···Cn− 1 C

∗

n

∣

∣+(r+n)

∣

∣C

1 C 2 ···Cn− 1 Cn+1

∣

∣

=Vn−(r+n)T

(n+1,r)

n+1,n

. (4.9.14)

Hence,

T

′

+(r+n)T

(n+1,r)

n+1,n

=

n

∑

j=1

Vj

=

n

∑

j=1

(i−1)

n

∑

j=1

ψr+i+j− 1 Tij

=−T

(n+1,r)

n+1,n

n

∑

i=2

(i−1)δin

=−(n−1)T

(n+1,r)

n+1,n.

The theorem follows.