Page 132 - Determinants and Their Applications in Mathematical Physics

4.9 Hankelians 2 117

2. The Yamazaki–Hori determinant A n is defined as follows:
A n = |φ m | n , 0 ≤ m ≤ 2n − 2,

where
1 2 2 m+1 2 2 m+1 2 2
φ m = p (x − 1) + q (y − 1) , p + q =1.
m +1
Let
B n = |ψ m | n , 0 ≤ m ≤ 2n − 2,
where

ψ m = φ m .
2
2 m+1
(x − y )
Prove that
= mFψ m−1 ,
∂ψ m
∂x
where
2
2x(y − 1)
F = − 2 2 2 .
(x − y )
Hence, prove that

(n)
= FB 11 ,
∂B n
∂x
2
2
2
2

(x − y ) ∂A n =2x n A n − (y − 1)A (n) .
11
∂x
Deduce the corresponding formulas for ∂B n /∂y and ∂A n /∂y and hence
prove that A n satisfies the equation
x − 1 y − 1 2
2
2
z x + z y =2n z.
x y
3. If A n = |φ m | n ,0 ≤ m ≤ 2n−2, where φ m satisfies the Appell equation,
prove that
i1
a. (A ) = −φ A A j1 − (iA i+1,j + jA i,j+1 ), (i, j) =(n, n),
ij

0
n n n n n
b. (A nn 0 1n 2
) = −φ (A ) .
n n
4. Apply Theorem 4.33 and the Jacobi identity to prove that
2

A (n)
= φ 0 1n .
A n
A n−1 A n−1
Hence, prove (3b).