DLMF:18.27.E20 (Q5971)

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DLMF:18.27.E20
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    Statements

    Defining formula
    ∫ 0 ∞ S n ⁡ ( q 1 2 ⁢ x ; q ) ⁢ S m ⁡ ( q 1 2 ⁢ x ; q ) ⁢ exp ⁡ ( - ( ln ⁡ x ) 2 2 ⁢ ln ⁡ ( q - 1 ) ) ⁢ d x = 2 ⁢ π ⁢ q - 1 ⁢ ln ⁡ ( q - 1 ) q n ⁢ ( q ; q ) n ⁢ δ n , m . superscript subscript 0 Stieltjes-Wigert-polynomial-S 𝑛 superscript 𝑞 1 2 𝑥 𝑞 Stieltjes-Wigert-polynomial-S 𝑚 superscript 𝑞 1 2 𝑥 𝑞 superscript 𝑥 2 2 superscript 𝑞 1 𝑥 2 𝜋 superscript 𝑞 1 superscript 𝑞 1 superscript 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 Kronecker 𝑛 𝑚 {\displaystyle{\displaystyle\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q% \right)S_{m}\left(q^{\frac{1}{2}}x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln% \left(q^{-1}\right)}\right)\mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}% \right)}}{q^{n}\left(q;q\right)_{n}}\delta_{n,m}.}}
    0 references
    DLMF id
    DLMF:18.27.E20
    0 references
    Symbols used
    Q10768
    Defining formula
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5afdec
    0 references
    Stieltjes–Wigert polynomial
    Defining formula
    S n ⁡ ( x ; q ) Stieltjes-Wigert-polynomial-S 𝑛 𝑥 𝑞 {\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)}}
    xml-id
    C18.S27.E18.m2abdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    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.m2afdec
    0 references
    Q11728
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C18.S1.XMD3.m1rdec
    0 references
    Q11730
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C18.S1.XMD5.m1fdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1odec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1ndec
    0 references
     
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