DLMF:15.8.E1 (Q5058)

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DLMF:15.8.E1
No description defined

    Statements

    Defining formula
    𝐅 ⁑ ( a , b c ; z ) = ( 1 - z ) - a ⁒ 𝐅 ⁑ ( a , c - b c ; z z - 1 ) = ( 1 - z ) - b ⁒ 𝐅 ⁑ ( c - a , b c ; z z - 1 ) = ( 1 - z ) c - a - b ⁒ 𝐅 ⁑ ( c - a , c - b c ; z ) , scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 1 𝑧 π‘Ž scaled-hypergeometric-bold-F π‘Ž 𝑐 𝑏 𝑐 𝑧 𝑧 1 superscript 1 𝑧 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑏 𝑐 𝑧 𝑧 1 superscript 1 𝑧 𝑐 π‘Ž 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑐 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}% \mathbf{F}\left({a,c-b\atop c};\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left(% {c-a,b\atop c};\frac{z}{z-1}\right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c% };z\right),}}
    0 references
    DLMF id
    DLMF:15.8.E1
    0 references
    constraint
    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2adec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1dec
    0 references
    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1dec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1dec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1dec
    0 references
     
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