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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.6 Symmetry, Special Values, and Limits to Monomials

Contents

§18.6(i) Symmetry and Special Values

For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.

Laguerre

18.6.1 Ln(α)(0)=(α+1)nn!.
Table 18.6.1: Classical OP’s: symmetry and special values.
pn(x) pn(-x) pn(1) p2n(0) p2n+1(0)
Pn(α,β)(x) (-1)nPn(β,α)(x) (α+1)n/n!
Pn(α,α)(x) (-1)nPn(α,α)(x) (α+1)n/n! (-14)n(n+α+1)n/n! (-14)n(n+α+1)n+1/n!
Cn(λ)(x) (-1)nCn(λ)(x) (2λ)n/n! (-1)n(λ)n/n! 2(-1)n(λ)n+1/n!
Tn(x) (-1)nTn(x) 1 (-1)n (-1)n(2n+1)
Un(x) (-1)nUn(x) n+1 (-1)n (-1)n(2n+2)
Vn(x) (-1)nWn(x) 2n+1 (-1)n (-1)n(2n+2)
Wn(x) (-1)nVn(x) 1 (-1)n (-1)n(2n+2)
Pn(x) (-1)nPn(x) 1 (-1)n(12)n/n! 2(-1)n(12)n+1/n!
Hn(x) (-1)nHn(x) (-1)n(n+1)n 2(-1)n(n+1)n+1
Hen(x) (-1)nHen(x) (-12)n(n+1)n (-12)n(n+1)n+1

§18.6(ii) Limits to Monomials

18.6.2 limαPn(α,β)(x)Pn(α,β)(1) =(1+x2)n,
18.6.3 limβPn(α,β)(x)Pn(α,β)(-1) =(1-x2)n,
18.6.4 limλCn(λ)(x)Cn(λ)(1) =xn,
18.6.5 limαLn(α)(αx)Ln(α)(0) =(1-x)n.