DLMF:31.16.E1 (Q9132)

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DLMF:31.16.E1
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    Statements

    Defining formula
    𝐻𝑝 n , m ⁡ ( x ) ⁢ 𝐻𝑝 n , m ⁡ ( y ) = ∑ j = 0 n A j ⁢ sin 2 ⁢ j ⁡ θ ⁢ P n - j ( γ + δ + 2 ⁢ j - 1 , ϵ - 1 ) ⁡ ( cos ⁡ ( 2 ⁢ θ ) ) ⁢ P j ( δ - 1 , γ - 1 ) ⁡ ( cos ⁡ ( 2 ⁢ ϕ ) ) , Heun-polynomial-m 𝑛 𝑚 𝑎 subscript 𝑞 𝑛 𝑚 𝑛 𝛽 𝛾 𝛿 𝑥 Heun-polynomial-m 𝑛 𝑚 𝑎 subscript 𝑞 𝑛 𝑚 𝑛 𝛽 𝛾 𝛿 𝑦 superscript subscript 𝑗 0 𝑛 subscript 𝐴 𝑗 2 𝑗 𝜃 Jacobi-polynomial-P 𝛾 𝛿 2 𝑗 1 italic-ϵ 1 𝑛 𝑗 2 𝜃 Jacobi-polynomial-P 𝛿 1 𝛾 1 𝑗 2 italic-ϕ {\displaystyle{\displaystyle\mathit{Hp}_{n,m}\left(x\right)\mathit{Hp}_{n,m}% \left(y\right)=\sum_{j=0}^{n}A_{j}{\sin^{2j}}\theta\*P^{(\gamma+\delta+2j-1,% \epsilon-1)}_{n-j}\left(\cos\left(2\theta\right)\right)P^{(\delta-1,\gamma-1)}% _{j}\left(\cos\left(2\phi\right)\right),}}
    0 references
    DLMF id
    DLMF:31.16.E1
    0 references
    Symbols used
    Heun polynomials
    Defining formula
    𝐻𝑝 n , m ⁡ ( a , q n , m ; - n , β , γ , δ ; z ) Heun-polynomial-m 𝑛 𝑚 𝑎 subscript 𝑞 𝑛 𝑚 𝑛 𝛽 𝛾 𝛿 𝑧 {\displaystyle{\displaystyle\mathit{Hp}_{\NVar{n},\NVar{m}}\left(\NVar{a},% \NVar{q_{n,m}};\NVar{-n},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}% \right)}}
    xml-id
    C31.S5.E2.m2adec
    0 references
    Q11660
    Defining formula
    P n ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r2.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q12458
    Defining formula
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C31.S1.XMD10.m1dec
    0 references
    Q12459
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C31.S1.XMD11.m1dec
    0 references
    Q12460
    Defining formula
    ϵ italic-ϵ {\displaystyle{\displaystyle\epsilon}}
    xml-id
    C31.S1.XMD12.m1dec
    0 references
    Q12477
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C31.S1.XMD2.m1dec
    0 references
    Q12483
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C31.S1.XMD4.m1dec
    0 references
    Q12484
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C31.S1.XMD5.m1dec
    0 references
    Q12461
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C31.S1.XMD6.m1dec
    0 references
    Q12462
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C31.S1.XMD7.m1dec
    0 references
    Q12464
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C31.S1.XMD9.m1dec
    0 references
    Q12516
    Defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C31.S16.XMD1.m1dec
    0 references
    Q12517
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C31.S16.XMD2.m1dec
    0 references
    Q12518
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C31.S16.XMD3.m1dec
    0 references
    Q12519
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C31.S16.XMD4.m1dec
    0 references
    Q12520
    Defining formula
    A j subscript 𝐴 𝑗 {\displaystyle{\displaystyle A_{j}}}
    xml-id
    C31.S16.XMD5.m1dec
    0 references
     
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