DLMF:15.9.E21 (Q5111)

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DLMF:15.9.E21
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    Statements

    Defining formula
    𝐅 ⁑ ( a , 1 - a c ; z ) = ( - z 1 - z ) ( 1 - c ) / 2 ⁒ P - a 1 - c ⁑ ( 1 - 2 ⁒ z ) , scaled-hypergeometric-bold-F π‘Ž 1 π‘Ž 𝑐 𝑧 superscript 𝑧 1 𝑧 1 𝑐 2 Legendre-P-first-kind 1 𝑐 π‘Ž 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,1-a\atop c};z\right)=\left(% \frac{-z}{1-z}\right)^{\ifrac{(1-c)}{2}}\*P^{1-c}_{-a}\left(1-2z\right),}}
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    DLMF id
    DLMF:15.9.E21
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    constraint
    | ph ⁑ ( - z ) | < Ο€ phase 𝑧 πœ‹ {\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi}}
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    Symbols used
    Q11481
    Defining formula
    P Ξ½ ΞΌ ⁑ ( z ) Legendre-P-first-kind πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aedec
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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.m2aedec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1fdec
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    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1edec
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