DLMF:15.9.E19 (Q5109)

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DLMF:15.9.E19
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    Statements

    Defining formula
    𝐅 ⁑ ( a , b a - b + 1 ; z ) = z ( b - a ) / 2 ⁒ ( 1 - z ) - b ⁒ P - b b - a ⁑ ( 1 + z 1 - z ) , scaled-hypergeometric-bold-F π‘Ž 𝑏 π‘Ž 𝑏 1 𝑧 superscript 𝑧 𝑏 π‘Ž 2 superscript 1 𝑧 𝑏 Legendre-P-first-kind 𝑏 π‘Ž 𝑏 1 𝑧 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a-b+1};z\right)=z^{% \ifrac{(b-a)}{2}}(1-z)^{-b}\*P^{b-a}_{-b}\left(\frac{1+z}{1-z}\right),}}
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    DLMF id
    DLMF:15.9.E19
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    constraint
    | ph ⁑ z | < Ο€ phase 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\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.m1acdec
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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.m2addec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
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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.m2acdec
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