Determinants and Their Applications in Mathematical Physics

328 Appendix

A.8 Differences

Given a sequence{ur}, thenthh-difference ofu 0 is written as ∆

n

h

u 0 and

is defined as

∆

n

hu^0 =

n

∑

r=0

(

n

r

)

(−h)

n−r

ur

=

n

∑

r=0

(

n

r

)

(−h)

r

un−r.

The first few differences are

∆

0

hu^0 =u^0 ,

∆

1

hu^0 =u^1 −hu^0 ,

∆

2

h

u 0 =u 2 − 2 hu 1 +h

2

u 0 ,

∆

3

h

u 0 =u 3 − 3 hu 2 +3h

2

u 1 −h

3

u 0.

The inverse relation is

un=

n

∑

r=0

(

n

r

)

(∆

r

h

u 0 )h

n−r

,

which is an Appell polynomial withαr=∆
r
h
u 0. Simple differences are

obtained by puttingh= 1 and are denoted by ∆

r

u 0.

Example A.1. If

ur=x

r

,

then

∆

n

h

u 0 =(x−h)

n

.

The proof is elementary.

Example A.2. If

ur=

1

2 r+1

,r≥ 1 , 0

then

∆

n

u 0 =

(−1)

n

2

2 n

n!

2

(2n+ 1)!

.

Proof.

∆

n

u 0 =

n

∑

r=0

(−1)

n−r

(

n

r

)

ur

=(−1)

n

n

∑

r=0

(−1)

r

(

n

r

)

1

2 r+1

=(−1)

n

f(1),