DLMF:15.9.E20 (Q5110)

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DLMF:15.9.E20
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    Statements

    Defining formula
    𝐅 ⁑ ( a , b 1 2 ⁒ ( a + b + 1 ) ; z ) = ( - z ⁒ ( 1 - z ) ) ( 1 - a - b ) / 4 ⁒ P ( a - b - 1 ) / 2 ( 1 - a - b ) / 2 ⁑ ( 1 - 2 ⁒ z ) , scaled-hypergeometric-bold-F π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 𝑧 superscript 𝑧 1 𝑧 1 π‘Ž 𝑏 4 Legendre-P-first-kind 1 π‘Ž 𝑏 2 π‘Ž 𝑏 1 2 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}(a+b+1)};z% \right)=\left(-z(1-z)\right)^{\ifrac{(1-a-b)}{4}}\*P^{\ifrac{(1-a-b)}{2}}_{% \ifrac{(a-b-1)}{2}}\left(1-2z\right),}}
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    DLMF id
    DLMF:15.9.E20
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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.m1addec
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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.m2aedec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1addec
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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.m2addec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1edec
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    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1ddec
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    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1cdec
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