Determinants and Their Applications in Mathematical Physics

224 5. Further Determinant Theory

=

∞

∑

m=0

z

m

um+1

m!

∞

∑

r=0

z

r

Fr

r!

. (5.7.13)

Equating coefficients ofz

n

,

Fn+1

n!

=

n

∑

r=0

ur+1Fn−r

r!(n−r)!

,

Fn+1=

n

∑

r=0

(

n

r

)

ur+1Fn−r. (5.7.14)

This recurrence relation inFnis identical in form to the recurrence relation

inEngiven in (5.7.7). Furthermore,

E 1 =F 1 =u 1 ,

E 2 =F 2 =u

2

1

+u 2.

Hence,

En=Fn

which proves the theorem.

Second proof.Express the lemma in the form

∞

∑

i=0

z

i

i!

H

i

(f, g)=f(x+z)g(x−z). (5.7.15)

Hence,

H

i

(f, g)=

[

D

i

z

{f(x+z)g(x−z)}

]

z=0

. (5.7.16)

Put

f(x)=e

F(x)

,

g(x)=e

G(x)

,

w=F(x+z)+G(x−z).

Then,

H

i

(e

F

,e

G

)=

[

D

i

z(e

w

)

]

z=0

=

[

D

i− 1

z

(e

w

wz)

]

z=0

=

i− 1

∑

j=0

(

i− 1

j

)

[

D

i−j

z

(w)D

j

z

(e

w

)

]

z=0

=

i− 1

∑

j=0

(

i− 1

j

)

ψi−jH

j

(e

F

,e

G

),i≥ 1 , (5.7.17)

where

ψr=

[

D

r

z

(w)

]

z=0