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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.9 Recurrence Relations and Derivatives

Contents

§18.9(i) Recurrence Relations

18.9.1 pn+1(x)=(Anx+Bn)pn(x)-Cnpn-1(x),

with initial values p0(x)=1 and p1(x)=A0x+B0.

For pn(x)=Pn(α,β)(x),

18.9.2 An =(2n+α+β+1)(2n+α+β+2)2(n+1)(n+α+β+1),
Bn =(α2-β2)(2n+α+β+1)2(n+1)(n+α+β+1)(2n+α+β),
Cn =(n+α)(n+β)(2n+α+β+2)(n+1)(n+α+β+1)(2n+α+β).

A0 and B0 have to be understood for α+β=0 or -1 by continuity in α and β, that is, A0=12(α+β)+1 and B0=12(α-β).

For the other classical OP’s see Table 18.9.1; compare also §18.2(iv).

Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).
pn(x) An Bn Cn
Cn(λ)(x) 2(n+λ)n+1 0 n+2λ-1n+1
Tn(x) 2-δn,0 0 1
Un(x) 2 0 1
Tn*(x) 4-2δn,0 -2+δn,0 1
Un*(x) 4 -2 1
Pn(x) 2n+1n+1 0 nn+1
Pn*(x) 4n+2n+1 -2n+1n+1 nn+1
Ln(α)(x) -1n+1 2n+α+1n+1 n+αn+1
Hn(x) 2 0 2n
Hen(x) 1 0 n

§18.9(ii) Contiguous Relations in the Parameters and the Degree

Jacobi

18.9.3 Pn(α,β-1)(x)-Pn(α-1,β)(x)=Pn-1(α,β)(x),
18.9.4 (1-x)Pn(α+1,β)(x)+(1+x)Pn(α,β+1)(x)=2Pn(α,β)(x).
18.9.5 (2n+α+β+1)Pn(α,β)(x)=(n+α+β+1)Pn(α,β+1)(x)+(n+α)Pn-1(α,β+1)(x),
18.9.6 (n+12α+12β+1)(1+x)Pn(α,β+1)(x)=(n+1)Pn+1(α,β)(x)+(n+β+1)Pn(α,β)(x),

and a similar pair to (18.9.5) and (18.9.6) by symmetry; compare the second row in Table 18.6.1.

Ultraspherical

18.9.7 (n+λ)Cn(λ)(x) =λ(Cn(λ+1)(x)-Cn-2(λ+1)(x)),
18.9.8 4λ(n+λ+1)(1-x2)Cn(λ+1)(x) =-(n+1)(n+2)Cn+2(λ)(x)+(n+2λ)(n+2λ+1)Cn(λ)(x).

Chebyshev

Laguerre

18.9.13 Ln(α)(x) =Ln(α+1)(x)-Ln-1(α+1)(x),
18.9.14 xLn(α+1)(x) =-(n+1)Ln+1(α)(x)+(n+α+1)Ln(α)(x).

§18.9(iii) Derivatives

Jacobi

18.9.15 ddxPn(α,β)(x)=12(n+α+β+1)Pn-1(α+1,β+1)(x),
18.9.16 ddx((1-x)α(1+x)βPn(α,β)(x))=-2(n+1)(1-x)α-1(1+x)β-1Pn+1(α-1,β-1)(x).
18.9.17 (2n+α+β)(1-x2)ddxPn(α,β)(x)=n(α-β-(2n+α+β)x)Pn(α,β)(x)+2(n+α)(n+β)Pn-1(α,β)(x),
18.9.18 (2n+α+β+2)(1-x2)ddxPn(α,β)(x)=(n+α+β+1)(α-β+(2n+α+β+2)x)Pn(α,β)(x)-2(n+1)(n+α+β+1)Pn+1(α,β)(x).

Ultraspherical

18.9.19 ddxCn(λ)(x)=2λCn-1(λ+1)(x),
18.9.20 ddx((1-x2)λ-12Cn(λ)(x))=-(n+1)(n+2λ-1)2(λ-1)(1-x2)λ-32Cn+1(λ-1)(x).

Chebyshev

Laguerre

Hermite

18.9.25 ddxHn(x) =2nHn-1(x),
18.9.26 ddx(e-x2Hn(x)) =-e-x2Hn+1(x).
18.9.27 ddxHen(x) =nHen-1(x),
18.9.28 ddx(e-12x2Hen(x)) =-e-12x2Hen+1(x).