r(号+ Pn(0) (E155) Pn"(x)=(-1) (-m) P(x) P(x)= ∑m-6)(k92+(-x)+(-1+门]E157) (n+1-m)Rn+1(x)+(n+m)Rn1(x)=(2n+1)xBm(x) (E158) (1-x2)Ram(x)=(n+1)xRm(x)-(n-m+1)Rm+1(x) (E159) (2n+1)xRn(x)=(+1)Rn+1(x)+nRn-1(x) (E.160) (x2-1)R(x)=(n+1)Rn+1(x)-xRn(x) Rn+1(x)-Rn-1(x)=(2n+1)Rn(x) (E162) Integral representations √cosb-cos Pn(x) Addition formula Pn(cos y)= Pn(cos 0)Pn(cos 8)+ +2 (n-m)! (φ一中) cosy= cos e cos6′+ sin 0 sin6’cos(φ-φ) (E.166) (E.167) rrcosy n=0/> cosy= cos e cos6+ sin e sin 0 cos(φ-φ) (E.168) GIrl, Irll GIrl, Irll @2001 by CRC Press LLCPn(0) =   n 2 + 1 2  √π   n 2 + 1  cos nπ 2 (E.155) P−m n (x) = (−1) m (n − m)! (n + m)! Pm n (x) (E.156) Power series Pn(x) = n k=0 (−1)k (n + k)! (n − k)!(k!)22k+1  (1 − x) k + (−1) n(1 + x) k  (E.157) Recursion relationships (n + 1 − m)Rm n+1(x) + (n + m)Rm n−1(x) = (2n + 1)x Rm n (x) (E.158) (1 − x 2 )Rm n (x) = (n + 1)x Rm n (x) − (n − m + 1)Rm n+1(x) (E.159) (2n + 1)x Rn(x) = (n + 1)Rn+1(x) + n Rn−1(x) (E.160) (x 2 − 1)R n(x) = (n + 1)[Rn+1(x) − x Rn(x)] (E.161) R n+1(x) − R n−1(x) = (2n + 1)Rn(x) (E.162) Integral representations Pn(cos θ) = √2 π  π 0 sin  n + 1 2  u √cos θ − cos u du (E.163) Pn(x) = 1 π  π 0  x + (x 2 − 1) 1/2 cos θ n dθ (E.164) Addition formula Pn(cos γ) = Pn(cos θ)Pn(cos θ ) + + 2 n m=1 (n − m)! (n + m)! Pm n (cos θ)Pm n (cos θ ) cos m(φ − φ ), (E.165) cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.166) Summations 1 |r − r | = 1  r 2 + r2 − 2rr cos γ = ∞ n=0 r n < r n+1 > Pn(cos γ) (E.167) cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.168) r< = min  |r|, |r |  , r> = max  |r|, |r |  (E.169)