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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.9 Recurrence Relations and Derivatives

Contents
  1. §18.9(i) Recurrence Relations
  2. §18.9(ii) Contiguous Relations in the Parameters and the Degree
  3. §18.9(iii) Derivatives

§18.9(i) Recurrence Relations

18.9.1 pn+1(x)=(Anx+Bn)pn(x)Cnpn1(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 =(α2β2)(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(αβ).

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+2λ1n+1
Tn(x) 2δn,0 0 1
Un(x) 2 0 1
Tn*(x) 42δn,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,β1)(x)Pn(α1,β)(x)=Pn1(α,β)(x),
18.9.4 (1x)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)Pn1(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)Cn2(λ+1)(x)),
18.9.8 4λ(n+λ+1)(1x2)Cn(λ+1)(x) =(n+1)(n+2)Cn+2(λ)(x)+(n+2λ)(n+2λ+1)Cn(λ)(x).

Chebyshev

18.9.9 Tn(x) =12(Un(x)Un2(x)),
18.9.10 (1x2)Un(x) =12(Tn+2(x)Tn(x)).
18.9.11 Vn(x)+Vn1(x) =2Tn(x),
18.9.12 Tn+1(x)+Tn(x) =(1+x)Vn(x).

Laguerre

18.9.13 Ln(α)(x) =Ln(α+1)(x)Ln1(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)Pn1(α+1,β+1)(x),
18.9.16 ddx((1x)α(1+x)βPn(α,β)(x))=2(n+1)(1x)α1(1+x)β1Pn+1(α1,β1)(x).
18.9.17 (2n+α+β)(1x2)ddxPn(α,β)(x)=n(αβ(2n+α+β)x)Pn(α,β)(x)+2(n+α)(n+β)Pn1(α,β)(x),
18.9.18 (2n+α+β+2)(1x2)ddxPn(α,β)(x)=(n+α+β+1)(αβ+(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)=2λCn1(λ+1)(x),
18.9.20 ddx((1x2)λ12Cn(λ)(x))=(n+1)(n+2λ1)2(λ1)(1x2)λ32Cn+1(λ1)(x).

Chebyshev

18.9.21 ddxTn(x)=nUn1(x),
18.9.22 ddx((1x2)12Un(x))=(n+1)(1x2)12Tn+1(x).

Laguerre

18.9.23 ddxLn(α)(x) =Ln1(α+1)(x),
18.9.24 ddx(exxαLn(α)(x)) =(n+1)exxα1Ln+1(α1)(x).

Hermite

18.9.25 ddxHn(x) =2nHn1(x),
18.9.26 ddx(ex2Hn(x)) =ex2Hn+1(x).
18.9.27 ddx𝐻𝑒n(x) =n𝐻𝑒n1(x),
18.9.28 ddx(e12x2𝐻𝑒n(x)) =e12x2𝐻𝑒n+1(x).