Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 139

Qn=Qn(x)=

[

x

2(i+j−1)

i+j− 1

]

n

. (4.11.12)

BothKnandQnare Hankelians andQn(1) =Kn, the simple Hilbert

matrix.

Sn=Sn(x)=

[

vnix

2 j− 1

i+j− 1

]

n

, (4.11.13)

where thevniarev-numbers.

Hn=Hn(x, t)=Sn(x)+tIn

=

[

h

(n)

ij

]

n

,

where

h

(n)

ij

=

vnix

2 j− 1

i+j− 1

+δijt,

Hn=Hn(x,−t)=Sn(x)−tIn

=

[

h

(n)

ij

]

n

, (4.11.14)

where

h

(n)

ij (x, t)=h

(n)

ij

(x,−t),

H ̄

n(x,−t)=(−1)

n

Hn(−x, t). (4.11.15)

Theorem 4.43.

K

− 1

nQn=S

2

n.

Proof. Referring to (4.11.7) and applying the formula for the product

of two matrices,

K

− 1

n

Qn=

[

vnivnj

i+j− 1

]

n

[

x

2(i+j−1)

i+j− 1

]

n

=

[

n

∑

k=1

vnivnk

i+k− 1

x

2(k+j−1)

k+j− 1

]

n

=

[

n

∑

k=1

(

vnix

2 k− 1

i+k− 1

)(

vnkx

2 j− 1

k+j− 1

)

]

n

=S

2

n. 

Theorem 4.44.

Bn=KnHnHn,

where the symbols can be interpreted as matrices or determinants.

Proof. Applying Theorem 4.43,

Bn=Qn−t

2

Kn