Determinants and Their Applications in Mathematical Physics

222 5. Further Determinant Theory

Applying Taylor’s theorem again,

1

2

{φ(x+z)−φ(x−z)}=

∞

∑

n=0

z

2 n+1

D

2 n+1

(φ)

(2n+ 1)!

,

1

2

{ψ(x+z)+ψ(x−z)}=

∞

∑

n=0

z

2 n

D

2 n

(ψ)

(2n)!

. (5.7.2)

5.7.2 A Determinantal Identity...............

Define functionsφ,ψ,unand a HessenbergianEnas follows:

φ= log(fg),

ψ= log(f/g) (5.7.3)

u 2 n=D

2 n

(φ),

u 2 n+1=D

2 n+1

(ψ), (5.7.4)

En=|eij|n,

where

eij=











(

j− 1

i− 1

)

uj−i+1,j≥i,

− 1 ,j=i−1,

0 , otherwise.

(5.7.5)

It follows from (5.7.3) that

f=e

(φ+ψ)/ 2

,

g=e

(φ−ψ)/ 2

,

fg=e

φ

. (5.7.6)

Theorem.

H

n

(f, g)

fg

=En

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

u 1 u 2 u 3 u 4 ··· un− 1 un

− 1 u 1 2 u 2 3 u 3 ··· ···

(

n− 1

n− 2

)

un− 1

− 1 u 1 3 u 2 ··· ···

(

n− 1

n− 3

)

un− 2

− 1 u 1 ··· ··· ···

··· ··· ···

− 1 u 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

.

This identity was conjectured by one of the authors and proved by Cau-

drey in 1984. The correspondence was private. Two proofs are given below.

The first is essentially Caudrey’s but with additional detail.