Determinants and Their Applications in Mathematical Physics

A.8 Differences 329

where

f(x)=

n

∑

r=0

(−1)

r

(

n

r

)

x

2 r+1

2 r+1

,

f

′

(x)=

n

∑

r=0

(−1)

r

(

n

r

)

x

2 r

=(1−x

2

)

n

.

f(x)=

∫x

0

(1−t

2

)

n

dt,

f(1) =

∫ 1

0

(1−t

2

)

n

dt

=

∫π/ 2

0

cos

2 n+1

θdθ

=

Γ

(

1

2

)

Γ(n+1)

2Γ

(

n+

3

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

ur=

x

2 r+2

−c

r+1

,

then

∆

n

u 0 =

(x

2

−1)

n+1

−(−1)

n

(c−1)

n+1

.

Proof.

∆

n

u 0 =

n

∑

r=0

(−1)

n−r

(

n

r

)[

x

2 r+2

− 1

r+1

−

c− 1

r+1

]

=(−1)

n

[S(x)+(c−1)S(0)],

where

S(x)=

n

∑

r=0

(−1)

r

(

n

r

)

x

2 r+2

− 1

r+1

=

1

n+1

n

∑

r=0

(−1)

r

(

n+1

r+1

)

(x

2 r+2

−1)

=

1

n+1

n+1

∑

r=0

(−1)

r+1

(

n+1

r

)

(x

2 r

−1), (Ther= 0 term is zero)