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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.6 Symmetry, Special Values, and Limits to Monomials

  1. §18.6(i) Symmetry and Special Values
  2. §18.6(ii) 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) 1 (1)n (1)n(2n+2)
Wn(x) (1)nVn(x) 2n+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) =(1x2)n,
18.6.4 limλCn(λ)(x)Cn(λ)(1) =xn,
18.6.5 limαLn(α)(αx)Ln(α)(0) =(1x)n.