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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, normalizations, 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 exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , see Abramowitz and Stegun (1964, pp. 793–801). …The ultraspherical polynomials are in powers of x for n = 0 , 1 , , 6 . …
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: 18.9 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 ) .
4: 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 1 λ C n ( λ ) ( x ) = 2 n T n ( x ) , n 1 .
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: 18.12 Generating Functions
    Ultraspherical
    18.12.4 ( 1 - 2 x z + z 2 ) - λ = n = 0 C n ( λ ) ( x ) z n = n = 0 ( 2 λ ) n ( λ + 1 2 ) n P n ( λ - 1 2 , λ - 1 2 ) ( x ) z n , | z | < 1 .
    18.12.5 1 - x z ( 1 - 2 x z + z 2 ) λ + 1 = n = 0 n + 2 λ 2 λ C n ( λ ) ( x ) z n , | z | < 1 .
    18.12.6 Γ ( λ + 1 2 ) e z cos θ ( 1 2 z sin θ ) 1 2 - λ J λ - 1 2 ( z sin θ ) = n = 0 C n ( λ ) ( cos θ ) ( 2 λ ) n z n , 0 θ π .
    8: 18.14 Inequalities
    Ultraspherical
    18.14.4 | C n ( λ ) ( x ) | C n ( λ ) ( 1 ) = ( 2 λ ) n n ! , - 1 x 1 , λ > 0 .
    18.14.5 | C 2 m ( λ ) ( x ) | | C 2 m ( λ ) ( 0 ) | = | ( λ ) m m ! | , - 1 x 1 , - 1 2 < λ < 0 ,
    18.14.6 | C 2 m + 1 ( λ ) ( x ) | < - 2 ( λ ) m + 1 ( ( 2 m + 1 ) ( 2 λ + 2 m + 1 ) ) 1 2 m ! , - 1 x 1 , - 1 2 < λ < 0 .
    18.14.7 ( n + λ ) 1 - λ ( 1 - x 2 ) 1 2 λ | C n ( λ ) ( x ) | < 2 1 - λ Γ ( λ ) , - 1 x 1 , 0 < λ < 1 .
    9: 18.5 Explicit Representations
    Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
    p n ( x ) F ( x ) κ n
    C n ( λ ) ( x ) 1 - x 2 ( - 2 ) n ( λ + 1 2 ) n n ! ( 2 λ ) n
    18.5.10 C n ( λ ) ( x ) = = 0 n / 2 ( - 1 ) ( λ ) n - ! ( n - 2 ) ! ( 2 x ) n - 2 = ( 2 x ) n ( λ ) n n ! F 1 2 ( - 1 2 n , - 1 2 n + 1 2 1 - λ - n ; 1 x 2 ) ,
    18.5.11 C n ( λ ) ( cos θ ) = = 0 n ( λ ) ( λ ) n - ! ( n - ) ! cos ( ( n - 2 ) θ ) = e i n θ ( λ ) n n ! F 1 2 ( - n , λ 1 - λ - n ; e - 2 i θ ) .
    Similarly in the cases of the ultraspherical polynomials C n ( λ ) ( x ) and the Laguerre polynomials L n ( α ) ( x ) we assume that λ > - 1 2 , λ 0 , and α > - 1 , unless stated otherwise. …
    10: 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 ) .