DLMF:15.6.E9 (Q5047)

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DLMF:15.6.E9
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    Statements

    test
    𝐅 ⁡ ( a , b ; c ; z ) = ∫ 0 1 t d - 1 ⁢ ( 1 - t ) c - d - 1 ( 1 - z ⁢ t ) a + b - λ ⁢ 𝐅 ⁡ ( λ - a , λ - b d ; z ⁢ t ) ⁢ 𝐅 ⁡ ( a + b - λ , λ - d c - d ; ( 1 - t ) ⁢ z 1 - z ⁢ t ) ⁢ d t , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 0 1 superscript 𝑡 𝑑 1 superscript 1 𝑡 𝑐 𝑑 1 superscript 1 𝑧 𝑡 𝑎 𝑏 𝜆 scaled-hypergeometric-bold-F 𝜆 𝑎 𝜆 𝑏 𝑑 𝑧 𝑡 scaled-hypergeometric-bold-F 𝑎 𝑏 𝜆 𝜆 𝑑 𝑐 𝑑 1 𝑡 𝑧 1 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t% ^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b% \atop d};zt\right)\mathbf{F}\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)% z}{1-zt}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:15.6.E9
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    constraint
    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    λ ∈ ℂ 𝜆 {\displaystyle{\displaystyle\lambda\in\mathbb{C}}}
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    ℜ ⁡ c > ℜ ⁡ d > 0 𝑐 𝑑 0 {\displaystyle{\displaystyle\Re c>\Re d>0}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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    Q10923
    test
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
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    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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