Determinants and Their Applications in Mathematical Physics

4.9 Hankelians 2 117

2.The Yamazaki–Hori determinantAnis defined as follows:

An=|φm|n, 0 ≤m≤ 2 n− 2 ,

where

φm=

1

m+1

[

p

2

(x

2

−1)

m+1

+q

2

(y

2

−1)

m+1

]

,p

2

+q

2

=1.

Let

Bn=|ψm|n, 0 ≤m≤ 2 n− 2 ,

where

ψm=

φm

(x

2

−y

2

)

m+1

.

Prove that

∂ψm

∂x

=mF ψm− 1 ,

where

F=−

2 x(y

2

−1)

(x

2

−y

2

)

2

.

Hence, prove that

∂Bn

∂x

=FB

(n)

11 ,

(x

2

−y

2

)

∂An

∂x

=2x

[

n

2

An−(y

2

−1)A

(n)

11

]

.

Deduce the corresponding formulas for∂Bn/∂yand∂An/∂yand hence

prove thatAnsatisfies the equation

(

x

2

− 1

x

)

zx+

(

y

2

− 1

y

)

zy=2n

2

z.

3.IfAn=|φm|n,0≤m≤ 2 n−2, whereφmsatisfies the Appell equation,

prove that

a.(A

ij

n

)

′

=−φ

′

0

A

i 1

n

A

j 1

n

−(iA

i+1,j

n

+jA

i,j+1

n

), (i, j)=(n, n),

b.(A

nn

n

)

′

=−φ

′

0

(A

1 n

n

)

2

.

4.Apply Theorem 4.33 and the Jacobi identity to prove that

(

An

An− 1

)′

=φ

′

0

(

A

(n)

1 n

An− 1

) 2

.

Hence, prove (3b).