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DLMF:17.3.E5 - MaRDI portal

DLMF:17.3.E5 (Q6042)

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DLMF:17.3.E5
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    Statements

    DLMF defining formula
    cos q ⁡ ( x ) = 1 2 ⁢ ( e q ⁡ ( i ⁢ x ) + e q ⁡ ( - i ⁢ x ) ) = ∑ n = 0 ∞ ( 1 - q ) 2 ⁢ n ⁢ ( - 1 ) n ⁢ x 2 ⁢ n ( q ; q ) 2 ⁢ n , q-cos 𝑞 𝑥 1 2 q-exp 𝑞 𝑖 𝑥 q-exp 𝑞 𝑖 𝑥 superscript subscript 𝑛 0 superscript 1 𝑞 2 𝑛 superscript 1 𝑛 superscript 𝑥 2 𝑛 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 {\displaystyle{\displaystyle\mathrm{cos}_{q}\left(x\right)=\frac{1}{2}(e_{q}% \left(ix\right)+e_{q}\left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n}(-1)% ^{n}x^{2n}}{\left(q;q\right)_{2n}},}}
    0 references
    Symbols used
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q11269
    DLMF defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2addec
    0 references
    $$q$$ -exponential function
    DLMF defining formula
    e q ⁡ ( x ) q-exp 𝑞 𝑥 {\displaystyle{\displaystyle e_{\NVar{q}}\left(\NVar{x}\right)}}
    xml-id
    C17.S3.E1.m2abdec
    0 references
    complex base
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1ddec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1ddec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C17.S1.XMD8.m1ddec
    0 references

    Identifiers

     
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