Determinants and Their Applications in Mathematical Physics

5.7 The One-Variable Hirota Operator 223

Proof. First proof(Caudrey). The Hessenbergian satisfies the recurrence

relation (Section 4.6)

En+1=

n

∑

r=0

(

n

r

)

ur+1En−r. (5.7.7)

Let

Fn=

H

n

(f, g)

fg

,f=f(x),g=g(x),F 0 =1. (5.7.8)

The theorem will be proved by showing that Fn satisfies the same

recurrence relation asEnand has the same initial values.

Let

K=























e

zH

(f,g)

fg

∑∞

n=0

z

n

n!

H

n

(f,g)

fg

∑∞

n=0

z

n

Fn

n!

.

(5.7.9)

Then,

∂K

∂z

=

∞

∑

n=1

z

n− 1

Fn

(n−1)!

(5.7.10)

=

∞

∑

n=0

z

n

Fn+1

n!

. (5.7.11)

From the lemma and (5.7.6),

K=

f(x+z)g(x−z)

f(x)g(x)

= exp

[

1

2

{φ(x+z)+φ(x−z)

+ψ(x+z)−ψ(x−z)− 2 φ(x)}

]

. (5.7.12)

Differentiate with respect toz, refer to (5.7.11), note that

Dz(φ(x−z)) =−Dx(φ(x−z))

etc., and apply the Taylor relations (5.7.2) from the previous section. The

result is

∞

∑

n=0

z

n

Fn+1

n!

=D

[

1

2

{φ(x+z)−φ(x−z)+ψ(x+z)+ψ(x−z)}

]

K

=

[

∞

∑

n=0

z

2 n+1

D

2 n+2

(φ)

(2n+ 1)!

+

∞

∑

n=0

z

2 n

D

2 n+1

(ψ)

(2n)!

]

K

=

[

∞

∑

n=0

z

2 n+1

u 2 n+2

(2n+ 1)!

+

∞

∑

n=0

z

2 n

u 2 n+1

(2n)!

]

K