Special Functions - Heat Conduction: Mathematical Models and Analytical Solutions - page 441

Note that

d x m 1

d m

x 2 v =

d x m + 2 1

d m + 2

x 2 +

d m + 1 v
d x m + 1 ( −

m
1!

− − 2 x )

d m v
d x m ( −

m ( m − 1 )

+ 2 ) ,

2

d x m 2 xv =

d m d m + 1 v

d x m + 1 2 x

d m v
d x m 2

m
1!

+ .

Thus
1

x 2 u −

d m v
d x m .

u +[ − 2 x ( m + 1 ) n ( n + 1 ) − m ( m + 1 )] u = 0 , u = (A.31)

Now consider the function transformation

= 1

x 2

m
2 u

m
2 w

x 2

) − − =( − .

w or u 1

Thus

d u
d x =

2 d w 2 −

x 2 ) −

1 w

x 2 ) − mx ( 1 − +( 1 −

d x ,

2 − 2 1

x 2 w

d 2 u
d x 2 =

x 2 +( x 2 ) −

m ( 1 − − m + 2 )

2 mx 1 x 2 −

d x + 1

x 2 −

2 d 2 w 2 − 1 d w

m m

d x 2 .
Substituting them into Eq. (A.31) yields the associated Legendre equations
1

+ − −

n

x 2 w

x 2 w −

m 2

2 xw + − ( n + 1 ) − = 0 .

1 −

When n is a natural number, therefore, the solutions of Eq. (A.29) are

= 1 x 2 = 1 x 2

d m P n

( x )

2 u 2

w − − .

d x m

The solution of the associated Legendre equations

)= 1 x 2

d m P n ( x )

2

P n ( x − , m ≤ n , | x | < 1

d x m

is called the associated Legendre polynomial of degree n and order m (see Table A.6.
By substituting the Rodrigue expression of P n ( into the definition of P n ( x ) x ) ,we

obtain the Rodrigue expression of P n ( x )

2 n n ! 1

x 2

d x n + m x 2

d n + m

1 n

1

m
2

P n ( x )= − − .

n , P n ( It is clear that if m > x ) ≡ 0.