递推关系之 将1-2+=∑P()2两端对微商 2x+2t 2(1-2xt+t2 ∑P(n) l=0
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials 4í'X ò 1 √ 1 − 2xt + t 2 = X ∞ l=0 Pl(x)t l üàétû − 1 2 −2x + 2t (1 − 2xt + t 2 ) 3/2 = X ∞ l=0 lPl(x)t l−1 x − t (1 − 2xt + t 2 ) 1/2 = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 (x − t) X ∞ l=0 Pl(x)t l = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 C. S. Wu 1Ôù ¥¼ê()
递推关系之 将 √1-2rt+t2 ∑P(a)t两端对微商 2x+2t 2(1-2rt+2)32 ∑P( l=0 1-2m+1)=(1-2+)∑(2y-1 ∑P(=(1=2+)∑()
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials 4í'X ò 1 √ 1 − 2xt + t 2 = X ∞ l=0 Pl(x)t l üàétû − 1 2 −2x + 2t (1 − 2xt + t 2 ) 3/2 = X ∞ l=0 lPl(x)t l−1 x − t (1 − 2xt + t 2 ) 1/2 = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 (x − t) X ∞ l=0 Pl(x)t l = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 C. S. Wu 1Ôù ¥¼ê()
递推关系之 将 √1-2rt+t2 ∑P(a)t两端对微商 2x+2t 2(1-2rt+2)32 ∑P(n) l=0 (1-2n+)=(1-2nt+(2)、 ∑P(n1=(1-2n+1)∑P()索
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials 4í'X ò 1 √ 1 − 2xt + t 2 = X ∞ l=0 Pl(x)t l üàétû − 1 2 −2x + 2t (1 − 2xt + t 2 ) 3/2 = X ∞ l=0 lPl(x)t l−1 x − t (1 − 2xt + t 2 ) 1/2 = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 (x − t) X ∞ l=0 Pl(x)t l = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 C. S. Wu 1Ôù ¥¼ê()
递推关系之 (x-1)∑P(x)2=(1-2t+t2)∑P(a)-1 l=0 比较项的系数,有 =(1+1)P+1(x)-2xP(x)+(1-1)P=1(
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials 4í'X (x − t) X ∞ l=0 Pl(x)t l = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 't lXê§k xPl(x) − Pl−1(x) = (l + 1)Pl+1(x) − 2xlPl(x) + (l − 1)Pl−1(x) n= (2l + 1)xPl(x) = (l + 1)Pl+1(x) + lPl−1(x) C. S. Wu 1Ôù ¥¼ê()
递推关系之 (x-1)∑P(x)2=(1-2t+t2)∑P(a)-1 l=0 比较t项的系数,有 aP(a)-Pi-1(a) (+1)P1+1(x)-2rlP(x)+(-1)P1-1(x) 整理即得 +1)xP(x)=(+1)P+1(x)+7P-1(x)
Properties of Legendre Polynomials (cont.) Applications of Legendre Polynomials Generating Function Recurrence Formulas for Legendre Polynomials 4í'X (x − t) X ∞ l=0 Pl(x)t l = 1 − 2xt + t 2 X ∞ l=0 lPl(x)t l−1 't lXê§k xPl(x) − Pl−1(x) = (l + 1)Pl+1(x) − 2xlPl(x) + (l − 1)Pl−1(x) n= (2l + 1)xPl(x) = (l + 1)Pl+1(x) + lPl−1(x) C. S. Wu 1Ôù ¥¼ê()