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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,

18.5.1 Tn(x) =cos(nθ),
18.5.2 Un(x) =(sin(n+1)θ)/sinθ,
18.5.3 Vn(x) =(cos(n+12)θ)/cos(12θ),
18.5.4 Wn(x) =(sin(n+12)θ)/sin(12θ).

§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, 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) F(x) κn
Pn(α,β)(x) 1x2 (2)nn!
Cn(λ)(x) 1x2 (2)n(λ+12)nn!(2λ)n
Tn(x) 1x2 (2)n(12)n
Un(x) 1x2 (2)n(32)nn+1
Vn(x) 1x2 (2)n(12)n
Wn(x) 1x2 (2)n(32)n2n+1
Pn(x) 1x2 (2)nn!
Ln(α)(x) x n!
Hn(x) 1 (1)n
𝐻𝑒n(x) 1 (1)n

Related formula:

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

§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).

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.

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).