DLMF:15.11.E7 (Q5159)

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DLMF:15.11.E7
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    Statements

    Defining formula
    P ⁡ { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } = ( z - α z - γ ) a 1 ⁢ ( z - β z - γ ) b 1 ⁢ P ⁡ { 0 1 ∞ 0 0 a 1 + b 1 + c 1 ( z - α ) ⁢ ( β - γ ) ( z - γ ) ⁢ ( β - α ) a 2 - a 1 b 2 - b 1 a 1 + b 1 + c 2 } . Riemann-P-symbol 𝛼 𝛽 𝛾 absent subscript 𝑎 1 subscript 𝑏 1 subscript 𝑐 1 𝑧 subscript 𝑎 2 subscript 𝑏 2 subscript 𝑐 2 absent superscript 𝑧 𝛼 𝑧 𝛾 subscript 𝑎 1 superscript 𝑧 𝛽 𝑧 𝛾 subscript 𝑏 1 Riemann-P-symbol 0 1 absent 0 0 subscript 𝑎 1 subscript 𝑏 1 subscript 𝑐 1 𝑧 𝛼 𝛽 𝛾 𝑧 𝛾 𝛽 𝛼 subscript 𝑎 2 subscript 𝑎 1 subscript 𝑏 2 subscript 𝑏 1 subscript 𝑎 1 subscript 𝑏 1 subscript 𝑐 2 absent {\displaystyle{\displaystyle P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=\left(\frac{z-\alpha}{z-\gamma}\right)^{a_{1}}% \left(\frac{z-\beta}{z-\gamma}\right)^{b_{1}}P\begin{Bmatrix}0&1&\infty&\\ 0&0&a_{1}+b_{1}+c_{1}&\dfrac{(z-\alpha)(\beta-\gamma)}{(z-\gamma)(\beta-\alpha% )}\\ a_{2}-a_{1}&b_{2}-b_{1}&a_{1}+b_{1}+c_{2}&\end{Bmatrix}.}}
    0 references
    DLMF id
    DLMF:15.11.E7
    0 references
    Symbols used
    Riemann’s $$P$$ -symbol for solutions of the generalized hypergeometric differential equation
    Defining formula
    P ⁡ { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 } Riemann-P-symbol 𝛼 𝛽 𝛾 absent subscript 𝑎 1 subscript 𝑏 1 subscript 𝑐 1 𝑧 subscript 𝑎 2 subscript 𝑏 2 subscript 𝑐 2 absent {\displaystyle{\displaystyle P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}}}
    xml-id
    C15.S11.E3.m2acdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1edec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1edec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1edec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1edec
    0 references
    Q11677
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C15.S11.XMD10.m1adec
    0 references
    Q11678
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C15.S11.XMD11.m1adec
    0 references
     
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