DLMF:18.22.E8 (Q5876)

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DLMF:18.22.E8
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    Statements

    Defining formula
    ( n + 1 ) ⁢ p n + 1 ⁢ ( x ) = 2 ⁢ ( x ⁢ sin ⁡ ϕ + ( n + λ ) ⁢ cos ⁡ ϕ ) ⁢ p n ⁢ ( x ) - ( n + 2 ⁢ λ - 1 ) ⁢ p n - 1 ⁢ ( x ) . 𝑛 1 subscript 𝑝 𝑛 1 𝑥 2 𝑥 italic-ϕ 𝑛 𝜆 italic-ϕ subscript 𝑝 𝑛 𝑥 𝑛 2 𝜆 1 subscript 𝑝 𝑛 1 𝑥 {\displaystyle{\displaystyle(n+1)p_{n+1}(x)=2\left(x\sin\phi+(n+\lambda)\cos% \phi\right)p_{n}(x)-(n+2\lambda-1)p_{n-1}(x).}}
    0 references
    DLMF id
    DLMF:18.22.E8
    0 references
    Symbols used
    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
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1hdec
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    Q11731
    Defining formula
    p n ⁢ ( x ) subscript 𝑝 𝑛 𝑥 {\displaystyle{\displaystyle p_{n}(x)}}
    xml-id
    C18.S1.XMD9.m1ddec
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