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21: 30.2 Differential Equations
30.2.1 d d z ( ( 1 z 2 ) d w d z ) + ( λ + γ 2 ( 1 z 2 ) μ 2 1 z 2 ) w = 0 .
The equation contains three real parameters λ , γ 2 , and μ . …
30.2.2 d 2 g d t 2 + ( λ + 1 4 + γ 2 sin 2 t μ 2 1 4 sin 2 t ) g = 0 ,
With ζ = γ z Equation (30.2.1) changes to
30.2.4 ( ζ 2 γ 2 ) d 2 w d ζ 2 + 2 ζ d w d ζ + ( ζ 2 λ γ 2 γ 2 μ 2 ζ 2 γ 2 ) w = 0 .
22: 28.19 Expansions in Series of me ν + 2 n Functions
28.19.2 f ( z ) = n = f n me ν + 2 n ( z , q ) ,
28.19.3 f n = 1 π 0 π f ( z ) me ν + 2 n ( z , q ) d z .
28.19.4 e i ν z = n = c 2 n ν + 2 n ( q ) me ν + 2 n ( z , q ) ,
23: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
31.5.1 [ 0 a γ 0 0 P 1 Q 1 R 1 0 0 P 2 Q 2 R n 1 0 0 P n Q n ] ,
31.5.2 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) = H ( a , q n , m ; n , β , γ , δ ; z )
24: 14.34 Software
§14.34(ii) Legendre Functions: Real Argument and Parameters
  • Adams and Swarztrauber (1997). Integer parameters. Fortran.

  • Braithwaite (1973). Integer parameters. Fortran.

  • Delic (1979a). Integer parameters. Fortran.

  • Gil and Segura (1998). Integer parameters. Fortran.

  • 25: 15.1 Special Notation
    26: 31.9 Orthogonality
    31.9.2 ζ ( 1 + , 0 + , 1 , 0 ) t γ 1 ( 1 t ) δ 1 ( t a ) ϵ 1 w m ( t ) w k ( t ) d t = δ m , k θ m .
    31.9.3 θ m = ( 1 e 2 π i γ ) ( 1 e 2 π i δ ) ζ γ ( 1 ζ ) δ ( ζ a ) ϵ f 0 ( q , ζ ) f 1 ( q , ζ ) q 𝒲 { f 0 ( q , ζ ) , f 1 ( q , ζ ) } | q = q m ,
    31.9.6 ρ ( s , t ) = ( s t ) ( s t ) γ 1 ( ( s 1 ) ( t 1 ) ) δ 1 ( ( s a ) ( t a ) ) ϵ 1 ,
    27: 31.11 Expansions in Series of Hypergeometric Functions
    31.11.3 λ + μ = γ + δ 1 = α + β ϵ .
    31.11.6 K j = ( j + α μ 1 ) ( j + β μ 1 ) ( j + γ μ 1 ) ( j μ ) ( 2 j + λ μ 1 ) ( 2 j + λ μ 2 ) ,
    31.11.7 L j = a ( λ + j ) ( μ j ) q + ( j + α μ ) ( j + β μ ) ( j + γ μ ) ( j + λ ) ( 2 j + λ μ ) ( 2 j + λ μ + 1 ) + ( j α + λ ) ( j β + λ ) ( j γ + λ ) ( j μ ) ( 2 j + λ μ ) ( 2 j + λ μ 1 ) ,
    31.11.12 P j 5 = ( α ) j ( 1 γ + α ) j ( 1 + α β + ϵ ) 2 j z α j F 1 2 ( α + j , 1 γ + α + j 1 + α β + ϵ + 2 j ; 1 z ) ,
    In this case the accessory parameter q is a root of the continued-fraction equation …
    28: 30.12 Generalized and Coulomb Spheroidal Functions
    30.12.1 d d z ( ( 1 z 2 ) d w d z ) + ( λ + α z + γ 2 ( 1 z 2 ) μ 2 1 z 2 ) w = 0 ,
    30.12.2 d d z ( ( 1 z 2 ) d w d z ) + ( λ + γ 2 ( 1 z 2 ) α ( α + 1 ) z 2 μ 2 1 z 2 ) w = 0 ,
    29: 31.16 Mathematical Applications
    It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. …
    31.16.5 P j = ( ϵ j + n ) j ( β + j 1 ) ( γ + δ + j 2 ) ( γ + δ + 2 j 3 ) ( γ + δ + 2 j 2 ) ,
    31.16.6 Q j = a j ( j + γ + δ 1 ) q + ( j n ) ( j + β ) ( j + γ ) ( j + γ + δ 1 ) ( 2 j + γ + δ ) ( 2 j + γ + δ 1 ) + ( j + n + γ + δ 1 ) j ( j + δ 1 ) ( j β + γ + δ 1 ) ( 2 j + γ + δ 1 ) ( 2 j + γ + δ 2 ) ,
    30: 33.13 Complex Variable and Parameters
    §33.13 Complex Variable and Parameters
    33.13.1 C ( η ) = 2 e i σ ( η ) ( π η / 2 ) Γ ( + 1 i η ) / Γ ( 2 + 2 ) ,
    33.13.2 R = ( 2 + 1 ) C ( η ) / C 1 ( η ) .