Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

230

Legendre polynomials

Observing the structure of coefficients a s + 2 (5.26), where s = 0 , 1 , 2 ,... , we can see that if k in the initial equation is an integer, then some of the coefficients a s + 2 can be equal to zero. For example, for k = s we have a k + 2 = a k + 4 = ··· = 0. If k is an even number, then y 1 is reduced to a polynomial of power k . If k is an odd number, then y 2 is reduced to a polynomial of power k . From (5.26) it follows that

( s + 1 )( s + 2 ) ( k − s )( k + s + 1 )

a s = −

a s + 2 , s ≤ k − 2 .

(5.28)

From this relation, all coefficients different from zero can be determined in terms of a k , namely the coefficient next to the highest power of x in the polynomial. This coefficient remains arbitrary. For convenience, it can be defined by the relation a k =    1 , k = 0 , ( 2 k ) ! 2 k ( k ! ) 2 = 1 · 3 · 5 ··· ( 2 k − 1 ) k ! , k = 1 , 2 ,... (5.29) For a k defined in this way, we obtain from (5.28)

k ( k − 1 ) 2 ( 2 k − 1 )

k ( k − 1 )( 2 k ) ! 2 ( 2 k − 1 ) 2 k ( k ! ) 2

a k

a k = −

− 2 = −

=

( 2 k − 2 ) ! 2 k ( k − 1 ) ! ( k − 2 ) !

= −

.. .

.. .

( 2 k − 2 m ) ! 2 k m ! ( k − m ) ! ( k − 2 m ) ! .

1 ) m

a k

− 2 m =( −

Definition The polynomial defined by the relation

M ∑ m = 0

( 2 k − 2 m ) ! 2 k m ! ( k − m ) ! ( k − 2 m ) !

( − 1 ) m

x k − 2 m

P k ( x )=

(5.30)

is called Legendre polynomial of power k , where M = k / 2 or ( k − 1 ) / 2 is awhole number.

This polynomial represents the solution of Legendre differential equation (5.21). Let us write down some of these polynomials

P 0 = 1;

P 1 = x

1 2

1 2

( 3 x 2 − 1 ) ;

( 5 x 3 − 3 x )

P 2 =

P

3 =

1 8 ( 35 x 4 − 30 x 2 + 3 ) ; P

1 8 ( 63 x 5 − 70 x 3 + 15 x ) .

P 4 =

5 =

Their graphs are represented in Figure 5.1.