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Gegenbauer polynomials

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1: 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 .
2: 18.9 Recurrence Relations and Derivatives
Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).
p n ( x ) A n B n C n
C n ( λ ) ( x ) 2 ( n + λ ) n + 1 0 n + 2 λ - 1 n + 1
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 ) .
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 ) .
3: 18.1 Notation
  • Ultraspherical (or Gegenbauer): C n ( λ ) ( x ) .

  • 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
    4: 18.10 Integral Representations
    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 .
    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 ).
    5: 18.12 Generating Functions
    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 .
    6: 18.6 Symmetry, Special Values, and Limits to Monomials
    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 !
    18.6.4 lim λ C n ( λ ) ( x ) C n ( λ ) ( 1 ) = x n ,
    7: 18.14 Inequalities
    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 .
    8: 18.17 Integrals
    18.17.5 C n ( λ ) ( cos θ 1 ) C n ( λ ) ( 1 ) C n ( λ ) ( cos θ 2 ) C n ( λ ) ( 1 ) = Γ ( λ + 1 2 ) π 1 2 Γ ( λ ) 0 π C n ( λ ) ( cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos ϕ ) C n ( λ ) ( 1 ) ( sin ϕ ) 2 λ - 1 d ϕ , λ > 0 .
    18.17.12 Γ ( λ - μ ) C n ( λ - μ ) ( x - 1 2 ) x λ - μ + 1 2 n = x Γ ( λ ) C n ( λ ) ( y - 1 2 ) y λ + 1 2 n ( y - x ) μ - 1 Γ ( μ ) d y , λ > μ > 0 , x > 0 ,
    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 .
    18.17.17 0 1 ( 1 - x 2 ) λ - 1 2 C 2 n ( λ ) ( x ) cos ( x y ) d x = ( - 1 ) n π Γ ( 2 n + 2 λ ) J λ + 2 n ( y ) ( 2 n ) ! Γ ( λ ) ( 2 y ) λ ,
    18.17.18 0 1 ( 1 - x 2 ) λ - 1 2 C 2 n + 1 ( λ ) ( x ) sin ( x y ) d x = ( - 1 ) n π Γ ( 2 n + 2 λ + 1 ) J 2 n + λ + 1 ( y ) ( 2 n + 1 ) ! Γ ( λ ) ( 2 y ) λ .
    9: 18.18 Sums
    18.18.16 C n ( μ ) ( x ) = = 0 n / 2 λ + n - 2 λ ( μ ) n - ( λ + 1 ) n - ( μ - λ ) ! C n - 2 ( λ ) ( x ) ,
    18.18.17 ( 2 x ) n = n ! = 0 n / 2 λ + n - 2 λ 1 ( λ + 1 ) n - ! C n - 2 ( λ ) ( x ) .
    18.18.22 C m ( λ ) ( x ) C n ( λ ) ( x ) = = 0 min ( m , n ) ( m + n + λ - 2 ) ( m + n - 2 ) ! ( m + n + λ - ) ! ( m - ) ! ( n - ) ! ( λ ) ( λ ) m - ( λ ) n - ( 2 λ ) m + n - ( λ ) m + n - ( 2 λ ) m + n - 2 C m + n - 2 ( λ ) ( x ) .
    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: 18.3 Definitions
    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). …