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1: 18.3 Definitions
§18.3 Definitions
This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre.
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
Ultraspherical (Gegenbauer) C n ( λ ) ( x ) ( 1 , 1 ) ( 1 x 2 ) λ 1 2 2 1 2 λ π Γ ( n + 2 λ ) ( n + λ ) ( Γ ( λ ) ) 2 n ! 2 n ( λ ) n n ! 0 λ > 1 2 , λ 0
For explicit power series coefficients up to n = 12 for these polynomials and for coefficients up to n = 6 for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …
2: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. …
Table 18.6.1: Classical OP’s: symmetry and special values.
p n ( x ) p n ( x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
C n ( λ ) ( x ) ( 1 ) n C n ( λ ) ( x ) ( 2 λ ) n / n ! ( 1 ) n ( λ ) n / n ! 2 ( 1 ) n ( λ ) n + 1 / n !
§18.6(ii) Limits to Monomials
18.6.4 lim λ C n ( λ ) ( x ) C n ( λ ) ( 1 ) = x n ,
3: 18.7 Interrelations and Limit Relations
18.7.1 C n ( λ ) ( x ) = ( 2 λ ) n ( λ + 1 2 ) n P n ( λ 1 2 , λ 1 2 ) ( x ) ,
18.7.9 P n ( x ) = C n ( 1 2 ) ( x ) = P n ( 0 , 0 ) ( x ) .
18.7.15 C 2 n ( λ ) ( x ) = ( λ ) n ( 1 2 ) n P n ( λ 1 2 , 1 2 ) ( 2 x 2 1 ) ,
18.7.24 lim λ λ 1 2 n C n ( λ ) ( λ 1 2 x ) = H n ( x ) n ! .
18.7.25 lim λ 0 n + λ λ C n ( λ ) ( x ) = { 1 , n = 0 , 2 T n ( x ) , n = 1 , 2 , .
4: 18.9 Recurrence Relations and Derivatives
18.9.7 ( n + λ ) C n ( λ ) ( x ) = λ ( C n ( λ + 1 ) ( x ) C n 2 ( λ + 1 ) ( x ) ) ,
18.9.8 4 λ ( n + λ + 1 ) ( 1 x 2 ) C n ( λ + 1 ) ( x ) = ( n + 1 ) ( n + 2 ) C n + 2 ( λ ) ( x ) + ( n + 2 λ ) ( n + 2 λ + 1 ) C n ( λ ) ( x ) .
Ultraspherical
18.9.19 d d x C n ( λ ) ( x ) = 2 λ C n 1 ( λ + 1 ) ( x ) ,
18.9.20 d d x ( ( 1 x 2 ) λ 1 2 C n ( λ ) ( x ) ) = ( n + 1 ) ( n + 2 λ 1 ) 2 ( λ 1 ) ( 1 x 2 ) λ 3 2 C n + 1 ( λ 1 ) ( x ) .
5: 18.1 Notation
  • Ultraspherical (or Gegenbauer): C n ( λ ) ( x ) .

  • Continuous q -Ultraspherical: C n ( x ; β | q ) .

  • In Szegő (1975, §4.7) the ultraspherical polynomials C n ( λ ) ( x ) are denoted by P n ( λ ) ( x ) . The ultraspherical polynomials will not be considered for λ = 0 . They are defined in the literature by C 0 ( 0 ) ( x ) = 1 and …
    6: 18.10 Integral Representations
    Ultraspherical
    18.10.1 P n ( α , α ) ( cos θ ) P n ( α , α ) ( 1 ) = C n ( α + 1 2 ) ( cos θ ) C n ( α + 1 2 ) ( 1 ) = 2 α + 1 2 Γ ( α + 1 ) π 1 2 Γ ( α + 1 2 ) ( sin θ ) 2 α 0 θ cos ( ( n + α + 1 2 ) ϕ ) ( cos ϕ cos θ ) α + 1 2 d ϕ , 0 < θ < π , α > 1 2 .
    Ultraspherical
    18.10.4 P n ( α , α ) ( cos θ ) P n ( α , α ) ( 1 ) = C n ( α + 1 2 ) ( cos θ ) C n ( α + 1 2 ) ( 1 ) = Γ ( α + 1 ) π 1 2 Γ ( α + 1 2 ) 0 π ( cos θ + i sin θ cos ϕ ) n ( sin ϕ ) 2 α d ϕ , α > 1 2 .
    Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).
    p n ( x ) g 0 ( x ) g 1 ( z , x ) g 2 ( z , x ) c Conditions
    C n ( λ ) ( x ) 1 z 1 ( 1 2 x z + z 2 ) λ 0 e ± i θ outside C (where x = cos θ ).
    7: 37.4 Disk with Weight Function ( 1 x 2 y 2 ) α
    37.4.5 C k , n ( α + 1 2 ) ( x , y ) = C n k ( α + k + 1 ) ( x ) ( 1 x 2 ) 1 2 k C k ( α + 1 2 ) ( y 1 x 2 ) .
    There is also an orthogonal basis of 𝒱 n α consisting of polynomials C k , n ( α + 1 2 ) ( y , x ) ( k = 0 , 1 , , n ). … For α = 0 , by (18.7.4), C n ( 1 ) = U n , the Chebyshev polynomial of the second kind. …
    37.4.29 [ ( 1 x 2 y 2 ) D y y 2 ( α + 1 ) y D y ] C k , n ( α + 1 2 ) ( x , y ) = k ( k + 2 α + 1 ) C k , n ( α + 1 2 ) ( x , y ) ,
    8: 37.11 Spherical Harmonics
    37.11.15 Y ( ξ ) = const . P n ( 1 2 ( d 3 ) , 1 2 ( d 3 ) ) ( ξ 1 ) = const . C n ( 1 2 ( d 2 ) ) ( ξ 1 ) , ξ = ( ξ 1 , , ξ d ) 𝕊 d 1 .
    37.11.18 Y 𝐧 ( ξ ) = e ± i n d 1 θ d 1 j = 1 d 2 C n j n j + 1 ( 1 2 ( d j 1 ) + n j + 1 ) ( cos θ j ) ( sin θ j ) n j + 1 , n = n 1 n 2 n d 1 0
    37.11.27 𝐑 n ( ξ , η ) = j = 1 N n d Y j ( ξ ) Y j ( η ) ¯ = N n d C n ( 1 2 ( d 2 ) ) ( ξ , η ) C n ( 1 2 ( d 2 ) ) ( 1 ) = Z n ( 1 2 ( d 2 ) ) ( ξ , η ) , ξ , η 𝕊 d 1 .
    37.11.28 Z n ( λ ) ( x ) = 1 1 ( 1 t 2 ) λ 1 2 d t 1 1 C n ( λ ) ( t ) 2 ( 1 t 2 ) λ 1 2 d t C n ( λ ) ( x ) C n ( λ ) ( 1 ) = n + λ λ C n ( λ ) ( x ) .
    9: 18.18 Sums
    Ultraspherical
    Ultraspherical
    Ultraspherical
    18.18.29 = 0 n C ( λ ) ( x ) C n ( μ ) ( x ) = C n ( λ + μ ) ( x ) ,
    18.18.30 = 0 n + 2 λ 2 λ C ( λ ) ( x ) x n = C n ( λ + 1 ) ( x ) .
    10: 18.17 Integrals
    Ultraspherical
    Ultraspherical
    18.17.13 x 1 2 n ( x 1 ) λ + μ 1 2 Γ ( λ + μ + 1 2 ) C n ( λ + μ ) ( x 1 2 ) C n ( λ + μ ) ( 1 ) = 1 x y 1 2 n ( y 1 ) λ 1 2 Γ ( λ + 1 2 ) C n ( λ ) ( y 1 2 ) C n ( λ ) ( 1 ) ( x y ) μ 1 Γ ( μ ) d y , μ > 0 , x > 1 .
    Ultraspherical
    Ultraspherical