Page 345 - Determinants and Their Applications in Mathematical Physics

330 Appendix

n+1
n+1
1 n +1 n +1
= − (−1) r x 2r − (−1) r
n +1 r r
r=0 r=0
1 2 n+1
= − [(1 − x ) − 0]
n +1
1
S(0) = − .
n +1
The result follows. It is applied with c = 1 in Section 4.10.4 on a particular
case of the Yamazaki–Hori determinant.
Example A.4. If
∞

ψ m = r x ,
m r
r=1
then
∆ ψ 0 = xψ m .
m
ψ m is the generalized geometric series (Appendix A.6).
Proof.

m
m

(r − 1) m = (−1) m−s r .
s
s
s=0
Multiply both sides by x and sum over r from 1 to ∞. (In the sum on the
r
left, the first term is zero and can therefore be omitted.)
∞ ∞ m
m r−1 m
x (r − 1) x = x r (−1) m−s r ,
s
s
r=2 r=1 s=0
∞ m
∞
m

x s x = (−1) m−s r x ,
m s
s r
s
s=1 s=0 r=1

m
m
xψ m = (−1) m−s
s ψ s
s=0
=∆ ψ 0 .
m
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 ,...,x n ). (A.9.1)