Page 202 - Determinants and Their Applications in Mathematical Physics

5.3 The Matsuno Identities 187
3
B 2n (φ)= −H n−1 H ,
n
2
B 2n (φ, φ)= H 2 H . (5.2.30)
n−1 n
These identities arose by a by-product in a study of Littlewood’s
Diophantime approximation problem.
The negative sign in the third identity, which is not required in Cusick’s
notation, arises from the difference between the methods by which B n (φ)
and Cusick’s determinant T n are defined. Note that B 2n (φ, φ)isskew-
symmetric of even order and is therefore expected to be a perfect square.

Exercises

1. Prove that
(2n) (n) (n)
A = −H n H K n K ,
1,2n 1n 1n
(2n−1) (n) (n)
A = H n−1 H K n−1 K .
1,2n−1 1n 1n
(2n)
2. Let V n (φ) be the determinant obtained from A by replacing the last
1,2n
(2n−1)
row by R 2n (φ) and let W n (φ) be the determinant obtained from A 1,2n−1
by replacing the last row by R 2n−1 (φ). Prove that
(n) (n)
V n (φ)= −H n H K n−1 K ,
1n 1n
(n) (n−1)
W n (φ)= −H n−1 H K n−1 K .
1n 1,n−1
3. Prove that
(2n) i+1 (n−1)
A =(−1) Pf n Pf .
i,2n i
5.3 The Matsuno Identities

Some of the identities in this section appear in Appendix II in a book on
the bilinear transformation method by Y. Matsuno, but the proofs have
been modified.

5.3.1 A General Identity
Let

A n = |a ij | n ,
where
 u ij , j = i


a ij = x − n ! u ir ,j = i, (5.3.1)
r=1


r =i