Classical Orthogonal Polynomials A Review Johar M. Ashfaque
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The Hermite Polynomials Hermite’s differential equation is defined as a second order differential equation d2 y dy − 2x + 2ny = 0 dx2 dx
where n is a non-negative integer, the solutions to which are referred to as Hermite polynomials. Hermite polynomials are denoted by Hn (x). The generating function for Hermite polynomials is 2
e2tx−t =
∞ X Hn (x)tn n! n=0
The first few Hermite polynomials are H0 (x) = 1 H1 (x) = 2x H2 (x) = 4x2 − 2 H3 (x) = 8x3 − 12x. Hermite polynomials Hn , form a complete orthogonal set on the interval −∞ < x < ∞, with respect to 2 the weighting function e−x such that ( Z ∞ 2 0, if m 6= n, e−x Hm (x)Hn (x)dx = √ n 2 n! π, if m = n. −∞
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The Laguerre Polynomials Laguerre’s differential equation is defined as a second order differential equation x
d2 y dy + (1 − x) + ny = 0 dx2 dx
where n is a non-negative integer, the solutions to which are referred to as Laguerre polynomials. Laguerre polynomials are denoted by Ln (x). The generating function for Laguerre polynomials is ∞ X e−xt/(1−t) = Ln (x)tn . 1−t n=0
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The first few Laguerre polynomials are L0 (x) = 1 L1 (x) = −x + 1 1 L2 (x) = (x2 − 4x + 2) 2 1 L3 (x) = (−x3 + 9x2 − 18x + 6). 6 Laguerre polynomials Ln , form a complete orthogonal set on the interval 0 < x < ∞ with respect to the weight function e−x , such that ( Z ∞ 0, if m = 6 n, Lm (x)Ln (x)dx = 1, if m = n. 0 Proposition 2.1 The recurrence relation for Laguerre polynomials is (n + 1)Ln+1 (x) = (2n + 1 − x)Ln (x) − nLn−1 (x).
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The Legendre Polynomials Legendre’s differential equation is defined as a second order differential equation (1 − x2 )
d2 y dy − 2x + l(l + 1)y = 0 2 dx dx
where n is a non-negative integer, the solutions to which are referred to as Legendre polynomials. Legendre polynomials are denoted by Pl (x). The generating function for Legendre polynomials is 1 p
(1 − 2tx + t2 )
=
∞ X
Pl (x)tl ,
t < 1.
l=0
The first few Legendre polynomials are P0 (x) = 1 P1 (x) = x 1 P2 (x) = (3x2 − 1) 2 1 P3 (x) = (5x3 − 3x). 2 Legendre polynomials Pl , form a complete orthogonal set on the interval −1 ≤ x ≤ 1, such that ( Z 1 0, if l 6= l0 , 2 Pl (x)Pl0 (x)dx = δll0 = 2 0 2l + 1 −1 2l+1 , if l = l .
Proposition 3.1 The recurrence relation for Legendre polynomials is (l + 1)Pl+1 (x) = (2l + 1)xPl (x) − lPl−1 (x).
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