Page 344 - Determinants and Their Applications in Mathematical Physics

A.8 Differences 329

where

n 2r+1
n x
f(x)= (−1) r ,
r 2r +1
r=0

n
n

f (x)= (−1) r x 2r
r
r=0
=(1 − x ) .
2 n
,
x
f(x)= (1 − t ) dt,
2 n
0
, 1
f(1) = (1 − t ) dt
2 n
0
π/2
,
= cos 2n+1 θdθ
0
1

Γ Γ(n +1)
= 2 3 .
2Γ n +
2
The proof is completed by applying the Legendre duplication formula for
the Gamma function (Appendix A.1). This result is applied at the end of
Section 4.10.3 on bordered Yamazaki–Hori determinants.
Example A.3. If
x 2r+2 − c
u r = ,
r +1
then
2
(x − 1) n+1 − (−1) (c − 1)
n
∆ u 0 = .
n
n +1
Proof.

n 2r+2
n x − 1 c − 1
∆ u 0 = (−1) n−r −
n
r r +1 r +1
r=0
=(−1) [S(x)+(c − 1)S(0)],
n
where

2r+2
n
n
x − 1
S(x)= (−1) r
r r +1
r=0
1 n +1
2r+2
n
= (−1) r (x − 1)
n +1 r +1
r=0
n+1
1 r+1 n +1
= (−1) (x 2r − 1), (The r = 0 term is zero)
n +1 r
r=0