DLMF:29.15.E49 (Q8794)

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DLMF:29.15.E49
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    Statements

    Defining formula
    𝑐𝑑𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) = cn ⁑ ( z , k ) ⁒ dn ⁑ ( z , k ) ⁒ βˆ‘ p = 0 n D 2 ⁒ p + 1 ⁒ U 2 ⁒ p ⁑ ( sn ⁑ ( z , k ) ) , Lame-polynomial-cdE π‘š 2 𝑛 2 𝑧 superscript π‘˜ 2 Jacobi-elliptic-cn 𝑧 π‘˜ Jacobi-elliptic-dn 𝑧 π‘˜ superscript subscript 𝑝 0 𝑛 subscript 𝐷 2 𝑝 1 Chebyshev-polynomial-second-kind-U 2 𝑝 Jacobi-elliptic-sn 𝑧 π‘˜ {\displaystyle{\displaystyle\mathit{cdE}^{m}_{2n+2}\left(z,k^{2}\right)=% \operatorname{cn}\left(z,k\right)\operatorname{dn}\left(z,k\right)\sum_{p=0}^{% n}D_{2p+1}U_{2p}\left(\operatorname{sn}\left(z,k\right)\right),}}
    0 references
    DLMF id
    DLMF:29.15.E49
    0 references
    Symbols used
    Q11661
    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.m2abdec
    0 references
    Jacobian elliptic function
    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.m2abdec
    0 references
    Jacobian elliptic function
    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.m2afdec
    0 references
    Jacobian elliptic function
    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.m2afdec
    0 references
    LamΓ© polynomial
    Defining formula
    𝑐𝑑𝐸 2 ⁒ n + 2 m ⁑ ( z , k 2 ) Lame-polynomial-cdE π‘š 2 𝑛 2 𝑧 superscript π‘˜ 2 {\displaystyle{\displaystyle\mathit{cdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z% },\NVar{k^{2}}\right)}}
    xml-id
    C29.S12.E7.m2aadec
    0 references
    Q12357
    Defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C29.S1.XMD1.m1vdec
    0 references
    Q12367
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C29.S1.XMD2.m1andec
    0 references
    Q12359
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C29.S1.XMD3.m1addec
    0 references
    Q12348
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C29.S1.XMD6.m1ndec
    0 references
    Q12350
    Defining formula
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C29.S1.XMD8.m1zdec
    0 references
    Q12375
    Defining formula
    D 2 ⁒ p subscript 𝐷 2 𝑝 {\displaystyle{\displaystyle D_{2p}}}
    xml-id
    C29.S6.XMD13.m1hdec
    0 references
     
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