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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. …
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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
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►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.
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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 !
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§18.6(ii) Limits to Monomials
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18.6.4 lim λ C n ( λ ) ⁡ ( x ) C n ( λ ) ⁡ ( 1 ) = x n ,
3: 18.9 Recurrence Relations and Derivatives
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18.9.7 ( n + λ ) ⁢ C n ( λ ) ⁡ ( x ) = λ ⁢ ( C n ( λ + 1 ) ⁡ ( x ) C n 2 ( λ + 1 ) ⁡ ( x ) ) ,
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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 ) .
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Ultraspherical
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18.9.19 d d x ⁡ C n ( λ ) ⁡ ( x ) = 2 ⁢ λ ⁢ C n 1 ( λ + 1 ) ⁡ ( x ) ,
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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
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18.7.1 C n ( λ ) ⁡ ( x ) = ( 2 ⁢ λ ) n ( λ + 1 2 ) n ⁢ P n ( λ 1 2 , λ 1 2 ) ⁡ ( x ) ,
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18.7.9 P n ⁡ ( x ) = C n ( 1 2 ) ⁡ ( x ) = P n ( 0 , 0 ) ⁡ ( x ) .
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18.7.15 C 2 ⁢ n ( λ ) ⁡ ( x ) = ( λ ) n ( 1 2 ) n ⁢ P n ( λ 1 2 , 1 2 ) ⁡ ( 2 ⁢ x 2 1 ) ,
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18.7.24 lim λ λ 1 2 ⁢ n ⁢ C n ( λ ) ⁡ ( λ 1 2 ⁢ x ) = H n ⁡ ( x ) n ! .
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18.7.25 lim λ 0 1 λ ⁢ C n ( λ ) ⁡ ( x ) = 2 n ⁢ T n ⁡ ( x ) , n 1 .
5: 18.18 Sums
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Ultraspherical
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Ultraspherical
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Ultraspherical
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18.18.29 ℓ = 0 n C ℓ ( λ ) ⁡ ( x ) ⁢ C n ℓ ( μ ) ⁡ ( x ) = C n ( λ + μ ) ⁡ ( x ) .
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18.18.30 ℓ = 0 n ℓ + 2 ⁢ λ 2 ⁢ λ ⁢ C ℓ ( λ ) ⁡ ( x ) ⁢ x n ℓ = C n ( λ + 1 ) ⁡ ( x ) .
6: 18.1 Notation
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  • Ultraspherical (or Gegenbauer): C n ( λ ) ⁡ ( x ) .

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  • 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 …
    7: 18.10 Integral Representations
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    Ultraspherical
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    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 .
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    Ultraspherical
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    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 .
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    Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).
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    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 ).
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    8: 18.12 Generating Functions
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    Ultraspherical
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    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 .
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    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 .
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    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 .
    9: 18.14 Inequalities
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    Ultraspherical
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    18.14.4 | C n ( λ ) ⁡ ( x ) | C n ( λ ) ⁡ ( 1 ) = ( 2 ⁢ λ ) n n ! , 1 x 1 , λ > 0 .
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    18.14.5 | C 2 ⁢ m ( λ ) ⁡ ( x ) | | C 2 ⁢ m ( λ ) ⁡ ( 0 ) | = | ( λ ) m m ! | , 1 x 1 , 1 2 < λ < 0 ,
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    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 .
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    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 .
    10: 18.5 Explicit Representations
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    Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
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    p n ⁡ ( x ) F ⁡ ( x ) κ n
    C n ( λ ) ⁡ ( x ) 1 x 2 ( 2 ) n ⁢ ( λ + 1 2 ) n ⁢ n ! ( 2 ⁢ λ ) n
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    18.5.10 C n ( λ ) ⁡ ( x ) = ℓ = 0 n / 2 ( 1 ) ℓ ⁢ ( λ ) n ℓ ℓ ! ⁢ ( n 2 ⁢ ℓ ) ! ⁢ ( 2 ⁢ x ) n 2 ⁢ ℓ = ( 2 ⁢ x ) n ⁢ ( λ ) n n ! ⁢ F 1 2 ⁡ ( 1 2 ⁢ n , 1 2 ⁢ n + 1 2 1 λ n ; 1 x 2 ) ,
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    18.5.11 C n ( λ ) ⁡ ( cos ⁡ θ ) = ℓ = 0 n ( λ ) ℓ ⁢ ( λ ) n ℓ ℓ ! ⁢ ( n ℓ ) ! ⁢ cos ⁡ ( ( n 2 ⁢ ℓ ) ⁢ 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. …