Determinants and Their Applications in Mathematical Physics

330 Appendix

=−

1

n+1

[

n+1

∑

r=0

(−1)

r

(

n+1

r

)

x

2 r

−

n+1

∑

r=0

(−1)

r

(

n+1

r

)

]

=−

1

n+1

[(1−x

2

)

n+1

−0]

S(0) =−

1

n+1

.

The result follows. It is applied withc= 1 in Section 4.10.4 on a particular

case of the Yamazaki–Hori determinant.

Example A.4. If

ψm=

∞

∑

r=1

r

m

x

r

,

then

∆

m

ψ 0 =xψm.

ψmis the generalized geometric series (Appendix A.6).

Proof.

(r−1)

m

=

m

∑

s=0

(−1)

m−s

(

m

s

)

r

s

.

Multiply both sides byx

r

and sum overrfrom 1 to∞. (In the sum on the

left, the first term is zero and can therefore be omitted.)

x

∞

∑

r=2

(r−1)

m

x

r− 1

=

∞

∑

r=1

x

r

m

∑

s=0

(−1)

m−s

(

m

s

)

r

s

,

x

∞

∑

s=1

s

m

x

s

=

m

∑

s=0

(−1)

m−s

(

m

s

) ∞

∑

r=1

r

s

x

r

,

xψm=

m

∑

s=0

(−1)

m−s

(

m

s

)

ψs

=∆

m

ψ 0.

This result is applied in Section 5.1.2 to prove Lawden’s theorem.

A.9 The Euler and Modified Euler Theorems on

Homogeneous Functions

The two theorems which follow concern two distinct kinds of homogeneity

of the function

f=f(x 0 ,x 1 ,x 2 ,...,xn). (A.9.1)