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Ferrers functions

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11: 14.2 Differential Equations
§14.2(ii) Associated Legendre Equation
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations P ν 0 ( x ) = P ν ( x ) , Q ν 0 ( x ) = Q ν ( x ) , P ν 0 ( x ) = P ν ( x ) , Q ν 0 ( x ) = Q ν ( x ) , Q ν 0 ( x ) = Q ν ( x ) = Q ν ( x ) / Γ ( ν + 1 ) . … Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions P ν μ ( x ) and Q ν μ ( x ) lie in the interval ( - 1 , 1 ) , and the arguments of the functions P ν μ ( x ) , Q ν μ ( x ) , and Q ν μ ( x ) lie in the interval ( 1 , ) . …
§14.2(iv) Wronskians and Cross-Products
14.2.5 P ν + 1 μ ( x ) Q ν μ ( x ) - P ν μ ( x ) Q ν + 1 μ ( x ) = Γ ( ν + μ + 1 ) Γ ( ν - μ + 2 ) ,
12: 14.9 Connection Formulas
§14.9(i) Connections Between P ν ± μ ( x ) , P - ν - 1 ± μ ( x ) , Q ν ± μ ( x ) , Q - ν - 1 μ ( x )
14.9.3 P ν - m ( x ) = ( - 1 ) m Γ ( ν - m + 1 ) Γ ( ν + m + 1 ) P ν m ( x ) ,
14.9.4 Q ν - m ( x ) = ( - 1 ) m Γ ( ν - m + 1 ) Γ ( ν + m + 1 ) Q ν m ( x ) , ν m - 1 , m - 2 , .
§14.9(ii) Connections Between P ν ± μ ( ± x ) , Q ν - μ ( ± x ) , Q ν μ ( x )
14.9.9 2 Γ ( ν + μ + 1 ) Γ ( μ - ν ) Q ν μ ( x ) = - cos ( ν π ) P ν - μ ( x ) + cos ( μ π ) P ν - μ ( - x ) ,
13: 14.3 Definitions and Hypergeometric Representations
§14.3 Definitions and Hypergeometric Representations
Ferrers Function of the First Kind
Ferrers Function of the Second Kind
§14.3(iii) Alternative Hypergeometric Representations
14: 30.8 Expansions in Series of Ferrers Functions
§30.8 Expansions in Series of Ferrers Functions
where P n + 2 k m ( x ) is the Ferrers function of the first kind (§14.3(i)), R = 1 2 ( n - m ) , and the coefficients a n , k m ( γ 2 ) are given by …
30.8.9 Qs n m ( x , γ 2 ) = k = - - N - 1 ( - 1 ) k a n , k m ( γ 2 ) P n + 2 k m ( x ) + k = - N ( - 1 ) k a n , k m ( γ 2 ) Q n + 2 k m ( x ) ,
where P n m and Q n m are again the Ferrers functions and N = 1 2 ( n + m ) . …
15: 14.33 Tables
§14.33 Tables
  • Zhang and Jin (1996, Chapter 4) tabulates P n ( x ) for n = 2 ( 1 ) 5 , 10 , x = 0 ( .1 ) 1 , 7D; P n ( cos θ ) for n = 1 ( 1 ) 4 , 10 , θ = 0 ( 5 ) 90 , 8D; Q n ( x ) for n = 0 ( 1 ) 2 , 10 , x = 0 ( .1 ) 0.9 , 8S; Q n ( cos θ ) for n = 0 ( 1 ) 3 , 10 , θ = 0 ( 5 ) 90 , 8D; P n m ( x ) for m = 1 ( 1 ) 4 , n - m = 0 ( 1 ) 2 , n = 10 , x = 0 , 0.5 , 8S; Q n m ( x ) for m = 1 ( 1 ) 4 , n = 0 ( 1 ) 2 , 10 , 8S; P ν m ( cos θ ) for m = 0 ( 1 ) 3 , ν = 0 ( .25 ) 5 , θ = 0 ( 15 ) 90 , 5D; P n ( x ) for n = 2 ( 1 ) 5 , 10 , x = 1 ( 1 ) 10 , 7S; Q n ( x ) for n = 0 ( 1 ) 2 , 10 , x = 2 ( 1 ) 10 , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 ν -zeros of P ν m ( cos θ ) and of its derivative for m = 0 ( 1 ) 4 , θ = 10 , 30 , 150 .

  • Žurina and Karmazina (1964, 1965) tabulate the conical functions P - 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = - 0.9 ( .1 ) 0.9 , 7S; P - 1 2 + i τ ( x ) for τ = 0 ( .01 ) 50 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7D. Auxiliary tables are included to facilitate computation for larger values of τ when - 1 < x < 1 .

  • Žurina and Karmazina (1963) tabulates the conical functions P - 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = - 0.9 ( .1 ) 0.9 , 7S; P - 1 2 + i τ 1 ( x ) for τ = 0 ( .01 ) 25 , x = 1.1 ( .1 ) 2 ( .2 ) 5 ( .5 ) 10 ( 10 ) 60 , 7S. Auxiliary tables are included to assist computation for larger values of τ when - 1 < x < 1 .

  • 16: 14.12 Integral Representations
    §14.12(i) - 1 < x < 1
    14.12.1 P ν μ ( cos θ ) = 2 1 / 2 ( sin θ ) μ π 1 / 2 Γ ( 1 2 - μ ) 0 θ cos ( ( ν + 1 2 ) t ) ( cos t - cos θ ) μ + ( 1 / 2 ) d t , 0 < θ < π , μ < 1 2 .
    14.12.2 P ν - μ ( x ) = ( 1 - x 2 ) - μ / 2 Γ ( μ ) x 1 P ν ( t ) ( t - x ) μ - 1 d t , μ > 0 ;
    14.12.3 Q ν μ ( cos θ ) = π 1 / 2 Γ ( ν + μ + 1 ) ( sin θ ) μ 2 μ + 1 Γ ( μ + 1 2 ) Γ ( ν - μ + 1 ) ( 0 ( sinh t ) 2 μ ( cos θ + i sin θ cosh t ) ν + μ + 1 d t + 0 ( sinh t ) 2 μ ( cos θ - i sin θ cosh t ) ν + μ + 1 d t ) , 0 < θ < π , μ > - 1 2 , ν ± μ > - 1 .
    17: 14.13 Trigonometric Expansions
    §14.13 Trigonometric Expansions
    14.13.1 P ν μ ( cos θ ) = 2 μ + 1 ( sin θ ) μ π 1 / 2 k = 0 Γ ( ν + μ + k + 1 ) Γ ( ν + k + 3 2 ) ( μ + 1 2 ) k k ! sin ( ( ν + μ + 2 k + 1 ) θ ) ,
    14.13.2 Q ν μ ( cos θ ) = π 1 / 2 2 μ ( sin θ ) μ k = 0 Γ ( ν + μ + k + 1 ) Γ ( ν + k + 3 2 ) ( μ + 1 2 ) k k ! cos ( ( ν + μ + 2 k + 1 ) θ ) .
    14.13.3 P n ( cos θ ) = 2 2 n + 2 ( n ! ) 2 π ( 2 n + 1 ) ! k = 0 1 3 ( 2 k - 1 ) k ! ( n + 1 ) ( n + 2 ) ( n + k ) ( 2 n + 3 ) ( 2 n + 5 ) ( 2 n + 2 k + 1 ) sin ( ( n + 2 k + 1 ) θ ) ,
    14.13.4 Q n ( cos θ ) = 2 2 n + 1 ( n ! ) 2 ( 2 n + 1 ) ! k = 0 1 3 ( 2 k - 1 ) k ! ( n + 1 ) ( n + 2 ) ( n + k ) ( 2 n + 3 ) ( 2 n + 5 ) ( 2 n + 2 k + 1 ) cos ( ( n + 2 k + 1 ) θ ) ,
    18: 14.8 Behavior at Singularities
    §14.8 Behavior at Singularities
    14.8.1 P ν μ ( x ) 1 Γ ( 1 - μ ) ( 2 1 - x ) μ / 2 , μ 1 , 2 , 3 , ,
    14.8.3 Q ν ( x ) = 1 2 ln ( 2 1 - x ) - γ - ψ ( ν + 1 ) + O ( 1 - x ) , ν - 1 , - 2 , - 3 , ,
    14.8.4 Q ν μ ( x ) 1 2 cos ( μ π ) Γ ( μ ) ( 2 1 - x ) μ / 2 , μ 1 2 , 3 2 , 5 2 , ,
    14.8.6 Q ν - μ ( x ) Γ ( μ ) Γ ( ν - μ + 1 ) 2 Γ ( ν + μ + 1 ) ( 2 1 - x ) μ / 2 , ν ± μ - 1 , - 2 , - 3 , .
    19: 14.23 Values on the Cut
    14.23.5 Q ν μ ( x ) = 1 2 Γ ( ν + μ + 1 ) ( e - μ π i / 2 Q ν μ ( x + i 0 ) + e μ π i / 2 Q ν μ ( x - i 0 ) ) ,
    14.23.6 Q ν μ ( x ) = e μ π i / 2 Γ ( ν + μ + 1 ) Q ν μ ( x ± i 0 ) ± 1 2 π i e ± μ π i / 2 P ν μ ( x ± i 0 ) .
    20: 18.11 Relations to Other Functions
    Ultraspherical
    18.11.1 P n m ( x ) = ( 1 2 ) m ( - 2 ) m ( 1 - x 2 ) 1 2 m C n - m ( m + 1 2 ) ( x ) = ( n + 1 ) m ( - 2 ) - m ( 1 - x 2 ) 1 2 m P n - m ( m , m ) ( x ) , 0 m n .
    For the Ferrers function P n m ( x ) , see §14.3(i). …