Page 239 - Determinants and Their Applications in Mathematical Physics

224 5. Further Determinant Theory
∞ ∞
z u m+1 z F r
m r
= . (5.7.13)
m! r!
m=0 r=0
Equating coefficients of z ,
n
n
F n+1 u r+1 F n−r
= ,
n! r!(n − r)!
r=0
n
n

F n+1 = u r+1 F n−r . (5.7.14)
r
r=0
This recurrence relation in F n is identical in form to the recurrence relation
in E n given in (5.7.7). Furthermore,
E 1 = F 1 = u 1 ,
2
E 2 = F 2 = u + u 2 .
1
Hence,

which proves the theorem. E n = F n
Second proof. Express the lemma in the form
∞
z i
H (f, g)= f(x + z)g(x − z). (5.7.15)
i
i!
i=0
Hence,

H (f, g)= D {f(x + z)g(x − z)} . (5.7.16)
i
i
z=0
z
Put
f(x)= e F (x) ,
g(x)= e G(x) ,
w = F(x + z)+ G(x − z).
Then,

H (e ,e )= D (e )
i
G
i
F
w
z=0
z
i−1
= D (e w z )
w
z=0
z
i−1
i − 1

= D i−j (w)D (e )
j
w
j z z z=0
j=0
i−1
i − 1

= ψ i−j H (e ,e ), i ≥ 1, (5.7.17)
j
F
G
j
j=0
where

ψ r = D (w)
r
z=0
z