Page 131 - Determinants and Their Applications in Mathematical Physics

116 4. Particular Determinants

where P m is the Legendre polynomial, then φ m satisfies (4.9.3) with
2 −3/2
F =(1 − x )
and φ 0 = P 0 = 1, but φ m is not a polynomial. These relations are applied
in Section 4.12.1 to evaluate |P m | n .

Examples

1. If
1 m+1
k
φ m = b r {f(x)+ c r } ,
m +1
r=1
where k ! b r =0, b r and c r are independent of x, and k is arbitrary,
r=1
then
φ = mf (x)φ m−1 ,

m
k

φ 0 = b r c r = constant.
r=1
Hence, A = |φ m | n is independent of x.
2. If
1 m+1 m+1 m+1
φ m (x, ξ)= (ξ + x) − c(ξ − 1) +(c − 1)ξ ,
m +1
then
= mφ m−1 ,
∂φ m
∂ξ
φ 0 = x + c.
Hence, A is independent of ξ. This relation is applied in Section 4.11.4
on a nonlinear differential equation.

Exercises
2
1. Denote the three cube roots of unity by 1, ω, and ω , and letA = |φ m | n ,
0 ≤ m ≤ 2n − 2, where
1 m+1 m+1 2 2 m+1
a. φ m = (x + b + c) + ω(x + ωc) + ω (x + ω c) ,
3(m +1)
1 m+1 2 m+1 2 m+1
b. φ m = (x + b + c) + ω (x + ωc) + ω(x + ω c) ,
3(m +1)
1 m+2 2 m+2 2 m+2
c. φ m = (x + c) + ω (x + ωc) + ω(x + ω c) .
3(m + 1)(m +2)
Prove that φ m and hence also A is real in each case, and that in cases
(a) and (b), A is independent of x, but in case (c), A = cA 11 .