Determinants and Their Applications in Mathematical Physics

116 4. Particular Determinants

wherePmis the Legendre polynomial, thenφmsatisfies (4.9.3) with

F=(1−x

2

)

− 3 / 2

andφ 0 =P 0 = 1, butφmis not a polynomial. These relations are applied

in Section 4.12.1 to evaluate|Pm|n.

Examples

1.If

φm=

1

m+1

k

∑

r=1

br{f(x)+cr}

m+1

,

where

∑k

r=1

br=0,brandcrare independent ofx, andkis arbitrary,

then

φ

′

m=mf

′

(x)φm− 1 ,

φ 0 =

k

∑

r=1

brcr= constant.

Hence,A=|φm|nis independent ofx.

2.If

φm(x, ξ)=

1

m+1

[

(ξ+x)

m+1

−c(ξ−1)

m+1

+(c−1)ξ

m+1

]

,

then

∂φm

∂ξ

=mφm− 1 ,

φ 0 =x+c.

Hence,Ais independent ofξ. This relation is applied in Section 4.11.4

on a nonlinear differential equation.

Exercises

1.Denote the three cube roots of unity by 1,ω, andω

2

, and letA=|φm|n,

0 ≤m≤ 2 n−2, where

a.φm=

1

3(m+1)

[

(x+b+c)

m+1

+ω(x+ωc)

m+1

+ω

2

(x+ω

2

c)

m+1

]

,

b.φm=

1

3(m+1)

[

(x+b+c)

m+1

+ω

2

(x+ωc)

m+1

+ω(x+ω

2

c)

m+1

]

,

c.φm=

1

3(m+ 1)(m+2)

[

(x+c)

m+2

+ω

2

(x+ωc)

m+2

+ω(x+ω

2

c)

m+2

]

.

Prove thatφmand hence alsoAis real in each case, and that in cases

(a) and (b),Ais independent ofx, but in case (c),A

′

=cA 11.