DLMF:29.15.E50 (Q9487)

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DLMF:29.15.E50
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    Statements

    DLMF defining formula
    𝑠𝑐𝑑𝐸 2 ⁒ n + 3 m ⁑ ( z , k 2 ) = cn ⁑ ( z , k ) ⁒ dn ⁑ ( z , k ) ⁒ βˆ‘ p = 0 n D 2 ⁒ p + 2 ⁒ U 2 ⁒ p + 1 ⁑ ( sn ⁑ ( z , k ) ) . Lame-polynomial-scdE π‘š 2 𝑛 3 𝑧 superscript π‘˜ 2 Jacobi-elliptic-cn 𝑧 π‘˜ Jacobi-elliptic-dn 𝑧 π‘˜ superscript subscript 𝑝 0 𝑛 subscript 𝐷 2 𝑝 2 Chebyshev-polynomial-second-kind-U 2 𝑝 1 Jacobi-elliptic-sn 𝑧 π‘˜ {\displaystyle{\displaystyle\mathit{scdE}^{m}_{2n+3}\left(z,k^{2}\right)=% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)\sum_{p=0}^{% n}D_{2p+2}U_{2p+1}\left(\operatorname{sn}\left(z,k\right)\right).}}
    0 references
    Symbols used
    Chebyshev polynomial of the second kind
    DLMF defining formula
    U n ⁑ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 π‘₯ {\displaystyle{\displaystyle U_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r5.m2acdec
    0 references
    Jacobian elliptic function
    DLMF defining formula
    cn ⁑ ( z , k ) Jacobi-elliptic-cn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2acdec
    0 references
    Jacobian elliptic function
    DLMF defining formula
    dn ⁑ ( z , k ) Jacobi-elliptic-dn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2agdec
    0 references
    Jacobian elliptic function
    DLMF defining formula
    sn ⁑ ( z , k ) Jacobi-elliptic-sn 𝑧 π‘˜ {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2agdec
    0 references
    LamΓ© polynomial
    DLMF defining formula
    𝑠𝑐𝑑𝐸 2 ⁒ n + 3 m ⁑ ( z , k 2 ) Lame-polynomial-scdE π‘š 2 𝑛 3 𝑧 superscript π‘˜ 2 {\displaystyle{\displaystyle\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{% z},\NVar{k^{2}}\right)}}
    xml-id
    C29.S12.E8.m2aadec
    0 references
    nonnegative integer
    DLMF defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1wdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C29.S1.XMD2.m1aodec
    0 references
    nonnegative integer
    DLMF defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C29.S1.XMD3.m1aedec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1odec
    0 references
    real parameter
    DLMF defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1aadec
    0 references
    coefficents
    DLMF defining formula
    D 2 ⁒ p + 1 subscript 𝐷 2 𝑝 1 {\displaystyle{\displaystyle D_{2p+1}}}
    xml-id
    C29.S6.XMD10.m1dec
    0 references

    Identifiers

     
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