Page 154 - Determinants and Their Applications in Mathematical Physics

4.11 Hankelians 4 139

2(i+j−1)
x
Q n = Q n (x)= . (4.11.12)
i + j − 1
n
Both K n and Q n are Hankelians and Q n (1) = K n , the simple Hilbert
matrix.
2j−1
v ni x
S n = S n (x)= , (4.11.13)
i + j − 1
n
where the v ni are v-numbers.
H n = H n (x, t)= S n (x)+ tI n
(n)
= h ,
ij
n
where
v ni x 2j−1
(n)
h = + δ ij t,
i + j − 1
ij
H n = H n (x, −t)= S n (x) − tI n
(n)
= h , (4.11.14)
ij
n
where
(n) (n)
h (x, t)= h (x, −t),
ij ij
¯
H n (x, −t)=(−1) H n (−x, t). (4.11.15)
n
Theorem 4.43.
2
K −1 Q n = S .
n n
Proof. Referring to (4.11.7) and applying the formula for the product
of two matrices,
2(i+j−1)
x
−1
K Q n = v ni v nj
i + j − 1 i + j − 1
n
n n

x
n 2(k+j−1)

= v ni v nk
i + k − 1 k + j − 1
k=1
n

n 2k−1
2j−1
v ni x v nk x
=
i + k − 1 k + j − 1
k=1
n
2
= S .
n
Theorem 4.44.
B n = K n H n H n ,
where the symbols can be interpreted as matrices or determinants.
Proof. Applying Theorem 4.43,
2
B n = Q n − t K n