Determinants and Their Applications in Mathematical Physics

5.3 The Matsuno Identities 187

B 2 n(φ)=−Hn− 1 H

3

n

,

B 2 n(φ, φ)=H

2

n− 1

H

2

n

. (5.2.30)

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 whichBn(φ)

and Cusick’s determinantTn are defined. Note thatB 2 n(φ, φ)isskew-

symmetric of even order and is therefore expected to be a perfect square.

Exercises

1.Prove that

A

(2n)

1 , 2 n=−HnH

(n)

1 nKnK

(n)

1 n,

A

(2n−1)

1 , 2 n− 1

=Hn− 1 H

(n)

1 n

Kn− 1 K

(n)

1 n

2.LetVn(φ) be the determinant obtained fromA

(2n)

1 , 2 nby replacing the last

row byR 2 n(φ) and letWn(φ) be the determinant obtained fromA

(2n−1)

1 , 2 n− 1

by replacing the last row byR 2 n− 1 (φ). Prove that

Vn(φ)=−HnH

(n)

1 n

Kn− 1 K

(n)

1 n

,

Wn(φ)=−Hn− 1 H

(n)

1 nKn−^1 K

(n−1)

1 ,n− 1.

3.Prove that

A

(2n)

i, 2 n

=(−1)

i+1

PfnPf

(n−1)

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

An=|aij|n,

where

aij=











uij,j=i

x−

∑n

r=1

r=i

uir,j=i, (5.3.1)