()=2n f(5) 2 n+1 兀l 故由(7)立即可证: .x)2(2+m)( ()=-2n2 f、d2(8) 丌l 二、缔合 Legendre多项式的性质 1.递推公式: (1+1-m)Pm(x)-(21+1)xP"(x) (1+m)(x)=09)
( ) ( ) ( ) ( ) 1 ! 2 - n l n n f f z d i z x x p x + Q = ò 故由(7)立即可证： ( ) ( ) ( ) ( ) ( ) 2 2 2 * 1 1- ! -1 (8) 2 ! 2 - m l m l l l l m x l m P x d l i x x x p x + + + = ò 二、缔合Legendre多项式的性质 1.递推公式： ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 -1 1- - 2 1 0 9 m m l l m l l m P x l xP x l m P x + + + + + =
(1+1)P1(x)-(21+1)xP(x) +P1(x)=0(B) ()(+2(x)(21+)yP(x) m(21+1)m(x)+m(x)=0( 又(21+1)P(x)=P1(x)P1(x)
( ) ( ) ( ) ( ) ( ) ( ) 1 -1 1 - 2 1 0 l l l l P x l xP x lP x B + + + + = Q ( ) ( ) ( ) ( ) ( ) ( ) : 1 1 - 2 1 ( ) m m m m l l d B l P x l xP x dx + + + ( ) ( ) ( ) ( ) ( ) ( ) -1 -1 - 2 1 0 10 m m m l l l + P x + = lP x ( ) ( ) ( ) ( ) 1 -1 2 1 - l l l l P x P x P x + 又 + = ¢ ¢