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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.7 Interrelations and Limit Relations

Contents

§18.7(i) Linear Transformations

Ultraspherical and Jacobi

18.7.1 Cn(λ)(x) =(2λ)n(λ+12)nPn(λ-12,λ-12)(x),
18.7.2 Pn(α,α)(x) =(α+1)n(2α+1)nCn(α+12)(x).

Chebyshev, Ultraspherical, and Jacobi

18.7.3 Tn(x)=Pn(-12,-12)(x)/Pn(-12,-12)(1),
18.7.4 Un(x)=Cn(1)(x)=(n+1)Pn(12,12)(x)/Pn(12,12)(1),
18.7.5 Vn(x)=(2n+1)Pn(12,-12)(x)/Pn(12,-12)(1),
18.7.6 Wn(x)=Pn(-12,12)(x)/Pn(-12,12)(1).
18.7.7 Tn*(x) =Tn(2x-1),
18.7.8 Un*(x) =Un(2x-1).

See also (18.9.9)–(18.9.12).

Legendre, Ultraspherical, and Jacobi

Hermite

18.7.11 Hen(x) =2-12nHn(2-12x),
18.7.12 Hn(x) =212nHen(212x).

§18.7(ii) Quadratic Transformations

18.7.13 P2n(α,α)(x)P2n(α,α)(1) =Pn(α,-12)(2x2-1)Pn(α,-12)(1),
18.7.14 P2n+1(α,α)(x)P2n+1(α,α)(1) =xPn(α,12)(2x2-1)Pn(α,12)(1).
18.7.15 C2n(λ)(x) =(λ)n(12)nPn(λ-12,-12)(2x2-1),
18.7.16 C2n+1(λ)(x) =(λ)n+1(12)n+1xPn(λ-12,12)(2x2-1).
18.7.17 U2n(x) =Vn(2x2-1),
18.7.18 T2n+1(x) =xWn(2x2-1).
18.7.19 H2n(x) =(-1)n22nn!Ln(-12)(x2),
18.7.20 H2n+1(x) =(-1)n22n+1n!xLn(12)(x2).

§18.7(iii) Limit Relations

Jacobi Laguerre

18.7.21 limβPn(α,β)(1-(2x/β))=Ln(α)(x).
18.7.22 limαPn(α,β)((2x/α)-1)=(-1)nLn(β)(x).

Jacobi Hermite

18.7.23 limαα-12nPn(α,α)(α-12x)=Hn(x)2nn!.

Ultraspherical Hermite

18.7.24 limλλ-12nCn(λ)(λ-12x)=Hn(x)n!.
18.7.25 limλ01λCn(λ)(x)=2nTn(x),
n1.

Laguerre Hermite

18.7.26 limα(2α)12nLn(α)((2α)12x+α)=(-1)nn!Hn(x).

See Figure 18.21.1 for the Askey schematic representation of most of these limits.