Page 343 - Determinants and Their Applications in Mathematical Physics

328 Appendix
A.8 Differences

Given a sequence {u r }, the nth h-difference of u 0 is written as ∆ u 0 and
n
h
is defined as
n
n
∆ u 0 = (−h) n−r
n
r
h u r
r=0
n
n
= (−h) u n−r .
r
r
r=0
The first few differences are
0
∆ u 0 = u 0 ,
h
1
∆ u 0 = u 1 − hu 0 ,
h
2
2
∆ u 0 = u 2 − 2hu 1 + h u 0 ,
h
2
3
3
∆ u 0 = u 3 − 3hu 2 +3h u 1 − h u 0 .
h
The inverse relation is
n
n

u n = (∆ u 0 )h n−r ,
r
r h
r=0
which is an Appell polynomial with α r =∆ u 0 . Simple differences are
r
h
obtained by putting h = 1 and are denoted by ∆ u 0 .
r
Example A.1. If
u r = x ,
r
then
∆ u 0 =(x − h) .
n
n
h
The proof is elementary.
Example A.2. If
1
u r = , r ≥ 1, 0
2r +1
then
(−1) 2 n! 2
n 2n
∆ u 0 = .
n
(2n + 1)!
Proof.

n
n

∆ u 0 = (−1) n−r
n
r u r
r=0

n
n 1
=(−1) n (−1) r
r 2r +1
r=0
=(−1) f(1),
n