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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.6 Symmetry, Special Values, and Limits to Monomials


§18.6(i) Symmetry and Special Values

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


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.