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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

The notation of §18.2(iv) will be used.

First Form

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

Second Form

18.9.2_1 xpn(x)=anpn+1(x)+bnpn(x)+cnpn1(x)

with initial values p0(x)=1 and p1(x)=a01(xb0).

For pn(x)=Pn(α,β)(x),

18.9.2_2 an =2(n+1)(n+α+β+1)(2n+α+β+1)(2n+α+β+2),
bn =β2α2(2n+α+β)(2n+α+β+2),
cn =2(n+α)(n+β)(2n+α+β)(2n+α+β+1).

a0 and b0 have to be understood for α+β=0 or 1 by continuity in α and β, that is, a0=2/(α+β+2) and and b0=(βα)/(α+β+2).

For the other classical OP’s see Table 18.9.2.

Table 18.9.2: Classical OP’s: recurrence relations (18.9.2_1).
pn(x) an bn cn
Cn(λ)(x) n+12(n+λ) 0 n+2λ12(n+λ)
Tn(x) 12(1+δn,0) 0 12
Un(x) 12 0 12
Vn(x) 12 12δn,0 12
Wn(x) 12 12δn,0 12
Pn(x) n+12n+1 0 n2n+1
Ln(α)(x) n1 2n+α+1 nα
Hn(x) 12 0 n

For the monic versions of the classical OP’s the recurrence coefficients bn and cn (there written as αn and βn, respectively) are given in §3.5(vi). They imply the recurrence coefficients for the orthonormal versions of the classical OP’s as well, see again §3.5(vi).

§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).

Identities similar to (18.9.11) and (18.9.12) involving Wn(x) and Tn(x) can be obtained using rows 4 and 7 in Table 18.6.1.

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).

Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17).

§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).

Further n-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7).

Formula (18.9.15) is degree lowering, while it raises the parameters. Formula (18.9.16) is degree raising, while it lowers the parameters. The following three formulas change the degree but preserve the parameters, see (18.2.42)–(18.2.44) for similar formulas for more general OP’s.

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),

and the structure relation

18.9.18_5 (1x2)ddxPn(α,β)(x)=2n(n+1)(n+α+β+1)(2n+α+β+1)(2n+α+β+2)Pn+1(α,β)(x)+2n(n+α+β+1)(αβ)(2n+α+β)(2n+α+β+2)Pn(α,β)(x)+2(n+α)(n+β)(n+α+β+1)(2n+α+β)(2n+α+β+1)Pn1(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).

See also the differentiation formulas in (Erdélyi et al., 1953b, §10.9(15))).

Chebyshev

Laguerre

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

Further n-th derivative formulas relating two different Laguerre polynomials can be obtained from §13.3(ii) by substitution of (13.6.19).

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).