DLMF:17.3.E4 (Q5344)

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DLMF:17.3.E4
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    Statements

    Defining formula
    Sin q ⁡ ( x ) = 1 2 ⁢ i ⁢ ( E q ⁡ ( i ⁢ x ) - E q ⁡ ( - i ⁢ x ) ) = ∑ n = 0 ∞ ( 1 - q ) 2 ⁢ n + 1 ⁢ q n ⁢ ( 2 ⁢ n + 1 ) ⁢ ( - 1 ) n ⁢ x 2 ⁢ n + 1 ( q ; q ) 2 ⁢ n + 1 . q-Sin 𝑞 𝑥 1 2 𝑖 q-Exp 𝑞 𝑖 𝑥 q-Exp 𝑞 𝑖 𝑥 superscript subscript 𝑛 0 superscript 1 𝑞 2 𝑛 1 superscript 𝑞 𝑛 2 𝑛 1 superscript 1 𝑛 superscript 𝑥 2 𝑛 1 q-Pochhammer-symbol 𝑞 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\mathrm{Sin}_{q}\left(x\right)=\frac{1}{2i}(E_{q}% \left(ix\right)-E_{q}\left(-ix\right))=\sum_{n=0}^{\infty}\frac{(1-q)^{2n+1}q^% {n(2n+1)}(-1)^{n}x^{2n+1}}{\left(q;q\right)_{2n+1}}.}}
    0 references
    DLMF id
    DLMF:17.3.E4
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    Symbols used
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    $$q$$ -exponential function
    Defining formula
    E q ⁡ ( x ) q-Exp 𝑞 𝑥 {\displaystyle{\displaystyle E_{\NVar{q}}\left(\NVar{x}\right)}}
    xml-id
    C17.S3.E2.m2aadec
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    Q11298
    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.m2acdec
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    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1cdec
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    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1cdec
    0 references
    Q11719
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C17.S1.XMD8.m1cdec
    0 references
     
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