Page 237 - Determinants and Their Applications in Mathematical Physics

222 5. Further Determinant Theory

Applying Taylor’s theorem again,
∞ 2n+1 2n+1
z D (φ)
1 {φ(x + z) − φ(x − z)} = ,
2 (2n + 1)!
n=0
∞
2n
2n
z D (ψ)
1
2 {ψ(x + z)+ ψ(x − z)} = (2n)! . (5.7.2)
n=0
5.7.2 A Determinantal Identity
Define functions φ, ψ, u n and a Hessenbergian E n as follows:
φ = log(fg),
ψ = log(f/g) (5.7.3)
u 2n = D (φ),
2n
u 2n+1 = D 2n+1 (ψ), (5.7.4)
E n = |e ij | n ,
where

j − 1
 u j−i+1 ,j ≥ i,
i − 1

e ij = (5.7.5)
 −1, j = i − 1,

0, otherwise.
It follows from (5.7.3) that
f = e (φ+ψ)/2 ,
g = e (φ−ψ)/2 ,
fg = e . (5.7.6)
φ
Theorem.
H (f, g)
n
fg = E n

u 1 u 2 u 3 u 4 ··· u n−1
u n
n − 1

−1 u 1 2u 2 3u 3 ··· ···

n − 2 u n−1

n − 1

−1 u 1 3u 2 ··· ··· u n−2 .

n − 3
=

−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.