About the Project
18 Orthogonal PolynomialsClassical Orthogonal Polynomials

§18.5 Explicit Representations


§18.5(i) Trigonometric Functions


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) 1-x2 (-2)nn!
Cn(λ)(x) 1-x2 (-2)n(λ+12)nn!(2λ)n
Tn(x) 1-x2 (-2)n(12)n
Un(x) 1-x2 (-2)n(32)nn+1
Vn(x) 1-x2 (-2)n(12)n
Wn(x) 1-x2 (-2)n(32)n2n+1
Pn(x) 1-x2 (-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(x-α-1e-1/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.


18.5.7 Pn(α,β)(x)==0n(n+α+β+1)(α++1)n-!(n-)!(x-12)=(α+1)nn!F12(-n,n+α+β+1α+1;1-x2),
18.5.8 Pn(α,β)(x)=2-n=0n(n+α)(n+βn-)(x-1)n-(x+1)=(α+1)nn!(x+12)nF12(-n,-n-βα+1;x-1x+1),

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


18.5.9 Cn(λ)(x)=(2λ)nn!F12(-n,n+2λλ+12;1-x2),
18.5.10 Cn(λ)(x)==0n/2(-1)(λ)n-!(n-2)!(2x)n-2=(2x)n(λ)nn!F12(-12n,-12n+121-λ-n;1x2),
18.5.11 Cn(λ)(cosθ)==0n(λ)(λ)n-!(n-)!cos((n-2)θ)=einθ(λ)nn!F12(-n,λ1-λ-n;e-2iθ).



18.5.13 Hn(x)=n!=0n/2(-1)(2x)n-2!(n-2)!=(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


18.5.14 T0(x) =1,
T1(x) =x,
T2(x) =2x2-1,
T3(x) =4x3-3x,
T4(x) =8x4-8x2+1,
T5(x) =16x5-20x3+5x,
T6(x) =32x6-48x4+18x2-1.
18.5.15 U0(x) =1,
U1(x) =2x,
U2(x) =4x2-1,
U3(x) =8x3-4x,
U4(x) =16x4-12x2+1,
U5(x) =32x5-32x3+6x,
U6(x) =64x6-80x4+24x2-1.


18.5.16 P0(x) =1,
P1(x) =x,
P2(x) =32x2-12,
P3(x) =52x3-32x,
P4(x) =358x4-154x2+38,
P5(x) =638x5-354x3+158x,
P6(x) =23116x6-31516x4+10516x2-516.


18.5.17 L0(x) =1,
L1(x) =-x+1,
L2(x) =12x2-2x+1,
L3(x) =-16x3+32x2-3x+1,
L4(x) =124x4-23x3+3x2-4x+1,
L5(x) =-1120x5+524x4-53x3+5x2-5x+1,
L6(x) =1720x6-120x5+58x4-103x3+152x2-6x+1.


18.5.18 H0(x) =1,
H1(x) =2x,
H2(x) =4x2-2,
H3(x) =8x3-12x,
H4(x) =16x4-48x2+12,
H5(x) =32x5-160x3+120x,
H6(x) =64x6-480x4+720x2-120.
18.5.19 He0(x) =1,
He1(x) =x,
He2(x) =x2-1,
He3(x) =x3-3x,
He4(x) =x4-6x2+3,
He5(x) =x5-10x3+15x,
He6(x) =x6-15x4+45x2-15.

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).