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18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.5 Explicit Representations

Contents
  1. §18.5(i) Trigonometric Functions
  2. §18.5(ii) Rodrigues Formulas
  3. §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
  4. §18.5(iv) Numerical Coefficients

§18.5(i) Trigonometric Functions

Chebyshev

With x=cosθ=12(z+z1),

18.5.1 Tn(x) =cos(nθ)=12(zn+zn),
18.5.2 Un(x) =sin((n+1)θ)sinθ=zn+1zn1zz1,
18.5.3 Vn(x) =cos((n+12)θ)cos(12θ)=zn+1+znz+1,
18.5.4 Wn(x) =sin((n+12)θ)sin(12θ)=zn+1znz1.
18.5.4_5 inUn(12i)=Fn+1.

In (18.5.4_5) see §26.11 for the Fibonacci numbers Fn.

§18.5(ii) Rodrigues Formulas

18.5.5 pn(x)=1κnw(x)dndxn(w(x)(F(x))n).

In this equation w(x) is as in Table 18.3.1, (reproduced in Table 18.5.1), and F(x), κn are as in Table 18.5.1.

Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
pn(x) w(x) F(x) κn
Pn(α,β)(x) (1x)α(1+x)β 1x2 (2)nn!
Cn(λ)(x) (1x2)λ12 1x2 (2)n(λ+12)nn!(2λ)n
Tn(x) (1x2)12 1x2 (2)n(12)n
Un(x) (1x2)12 1x2 (2)n(32)nn+1
Vn(x) (1+x1x)12 1x2 (2)n(12)n
Wn(x) (1x1+x)12 1x2 (2)n(32)n2n+1
Pn(x) 1 1x2 (2)nn!
Ln(α)(x) exxα x n!
Hn(x) ex2 1 (1)n
𝐻𝑒n(x) e12x2 1 (1)n

Related formula:

18.5.6 Ln(α)(1x)=(1)nn!xn+α+1e1/xdndxn(xα1e1/x).

See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials.

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

For the definitions of F12, F11, and F02 see §16.2.

Jacobi

18.5.7 Pn(α,β)(x)==0n(n+α+β+1)(α++1)n!(n)!(x12)=(α+1)nn!F12(n,n+α+β+1α+1;1x2),
18.5.8 Pn(α,β)(x)=2n=0n(n+α)(n+βn)(x1)n(x+1)=(α+1)nn!(x+12)nF12(n,nβα+1;x1x+1),

and two similar formulas by symmetry; compare the second row in Table 18.6.1.

Ultraspherical

18.5.9 Cn(λ)(x)=(2λ)nn!F12(n,n+2λλ+12;1x2),
18.5.10 Cn(λ)(x)==0n/2(1)(λ)n!(n2)!(2x)n2=(2x)n(λ)nn!F12(12n,12n+121λn;1x2),
18.5.11 Cn(λ)(cosθ)==0n(λ)(λ)n!(n)!cos((n2)θ)=einθ(λ)nn!F12(n,λ1λn;e2iθ).

Chebyshev

18.5.11_1 Tn(x)=12n=0n/2(1)(n1)!!(n2)!(2x)n2=2n1xnF12(12n,12n+121n;1x2),
n1,
18.5.11_2 Tn(x)=F12(n,n12;1x2),
18.5.11_3 Un(x)==0n/2(1)(n)!!(n2)!(2x)n2=(2x)nF12(12n,12n+12n;1x2),
18.5.11_4 Un(x)=(n+1)F12(n,n+232;1x2).

Laguerre

Hermite

18.5.13 Hn(x)=n!=0n/2(1)(2x)n2!(n2)!=(2x)nF02(12n,12n+12;1x2).

For corresponding formulas for Chebyshev, Legendre, and the Hermite 𝐻𝑒n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11).

Note. The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of Pn(α,β)(x) when the conditions α>1 and β>1 are not satisfied. However, in these circumstances the orthogonality property (18.2.1) disappears. For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials Pn(α,β)(x) we assume throughout this chapter that α>1 and β>1, unless stated otherwise. Similarly in the cases of the ultraspherical polynomials Cn(λ)(x) and the Laguerre polynomials Ln(α)(x) we assume that λ>12,λ0, and α>1, unless stated otherwise.

§18.5(iv) Numerical Coefficients

Chebyshev

18.5.14 T0(x) =1,
T1(x) =x,
T2(x) =2x21,
T3(x) =4x33x,
T4(x) =8x48x2+1,
T5(x) =16x520x3+5x,
T6(x) =32x648x4+18x21.
18.5.15 U0(x) =1,
U1(x) =2x,
U2(x) =4x21,
U3(x) =8x34x,
U4(x) =16x412x2+1,
U5(x) =32x532x3+6x,
U6(x) =64x680x4+24x21.

Legendre

18.5.16 P0(x) =1,
P1(x) =x,
P2(x) =32x212,
P3(x) =52x332x,
P4(x) =358x4154x2+38,
P5(x) =638x5354x3+158x,
P6(x) =23116x631516x4+10516x2516.

Laguerre

18.5.17 L0(x) =1,
L1(x) =x+1,
L2(x) =12x22x+1,
L3(x) =16x3+32x23x+1,
L4(x) =124x423x3+3x24x+1,
L5(x) =1120x5+524x453x3+5x25x+1,
L6(x) =1720x6120x5+58x4103x3+152x26x+1.
18.5.17_5 L0(α)(x) =1,
L1(α)(x) =x+α+1,
L2(α)(x) =12x2(α+2)x+12(α+1)(α+2),
L3(α)(x) =16x3+12(α+3)x212(α+2)2x+16(α+1)3,
L4(α)(x) =124x416(α+4)x3+14(α+3)2x216(α+2)3x+124(α+1)4.

Hermite

18.5.18 H0(x) =1,
H1(x) =2x,
H2(x) =4x22,
H3(x) =8x312x,
H4(x) =16x448x2+12,
H5(x) =32x5160x3+120x,
H6(x) =64x6480x4+720x2120.
18.5.19 𝐻𝑒0(x) =1,
𝐻𝑒1(x) =x,
𝐻𝑒2(x) =x21,
𝐻𝑒3(x) =x33x,
𝐻𝑒4(x) =x46x2+3,
𝐻𝑒5(x) =x510x3+15x,
𝐻𝑒6(x) =x615x4+45x215.

For the corresponding polynomials of degrees 7 through 12 see Abramowitz and Stegun (1964, Tables 22.3, 22.5, 22.9, 22.10, 22.12).