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Gegenbauer function

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1: 14.3 Definitions and Hypergeometric Representations
§14.3(iv) Relations to Other Functions
In terms of the Gegenbauer function C α ( β ) ( x ) and the Jacobi function ϕ λ ( α , β ) ( t ) (§§15.9(iii), 15.9(ii)):
14.3.21 P ν μ ( x ) = 2 μ Γ ( 1 - 2 μ ) Γ ( ν + μ + 1 ) Γ ( ν - μ + 1 ) Γ ( 1 - μ ) ( 1 - x 2 ) μ / 2 C ν + μ ( 1 2 - μ ) ( x ) .
14.3.22 P ν μ ( x ) = 2 μ Γ ( 1 - 2 μ ) Γ ( ν + μ + 1 ) Γ ( ν - μ + 1 ) Γ ( 1 - μ ) ( x 2 - 1 ) μ / 2 C ν + μ ( 1 2 - μ ) ( x ) .
2: 15.9 Relations to Other Functions
Gegenbauer (or Ultraspherical)
§15.9(iii) Gegenbauer Function
15.9.15 C α ( λ ) ( z ) = Γ ( α + 2 λ ) Γ ( 2 λ ) Γ ( α + 1 ) F ( - α , α + 2 λ λ + 1 2 ; 1 - z 2 ) .
3: 18.8 Differential Equations
Table 18.8.1: Classical OP’s: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
f ( x ) A ( x ) B ( x ) C ( x ) λ n
C n ( λ ) ( x ) 1 - x 2 - ( 2 λ + 1 ) x 0 n ( n + 2 λ )
4: Errata
  • Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

    The Gegenbauer function C α ( λ ) ( z ) , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial C n ( λ ) ( z ) . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

  • 5: 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 .
    6: 18.12 Generating Functions
    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 .
    7: Bibliography D
  • L. Durand (1975) Nicholson-type Integrals for Products of Gegenbauer Functions and Related Topics. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 353–374. Math. Res. Center, Univ. Wisconsin, Publ. No. 35.
  • 8: 18.14 Inequalities
    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: 18.17 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 .
    10: 18.15 Asymptotic Approximations
    18.15.10 C n ( λ ) ( cos θ ) = 2 2 λ Γ ( λ + 1 2 ) π 1 2 Γ ( λ + 1 ) ( 2 λ ) n ( λ + 1 ) n ( m = 0 M - 1 ( λ ) m ( 1 - λ ) m m ! ( n + λ + 1 ) m cos θ n , m ( 2 sin θ ) m + λ + O ( 1 n M ) ) ,