DLMF:16.17.E3 (Q5279)

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DLMF:16.17.E3
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    Statements

    Defining formula
    A p , q , k m , n ⁢ ( z ) = ∏ ℓ = 1 ℓ ≠ k m Γ ⁡ ( b ℓ - b k ) ⁢ ∏ ℓ = 1 n Γ ⁡ ( 1 + b k - a ℓ ) ⁢ z b k / ( ∏ ℓ = m q - 1 Γ ⁡ ( 1 + b k - b ℓ + 1 ) ⁢ ∏ ℓ = n p - 1 Γ ⁡ ( a ℓ + 1 - b k ) ) . superscript subscript 𝐴 𝑝 𝑞 𝑘 𝑚 𝑛 𝑧 superscript subscript product ℓ 1 ℓ 𝑘 𝑚 Euler-Gamma subscript 𝑏 ℓ subscript 𝑏 𝑘 superscript subscript product ℓ 1 𝑛 Euler-Gamma 1 subscript 𝑏 𝑘 subscript 𝑎 ℓ superscript 𝑧 subscript 𝑏 𝑘 superscript subscript product ℓ 𝑚 𝑞 1 Euler-Gamma 1 subscript 𝑏 𝑘 subscript 𝑏 ℓ 1 superscript subscript product ℓ 𝑛 𝑝 1 Euler-Gamma subscript 𝑎 ℓ 1 subscript 𝑏 𝑘 {\displaystyle{\displaystyle A_{p,q,k}^{m,n}(z)=\ifrac{\prod\limits_{% \begin{subarray}{c}\ell=1\\ \ell\neq k\end{subarray}}^{m}\Gamma\left(b_{\ell}-b_{k}\right)\prod\limits_{% \ell=1}^{n}\Gamma\left(1+b_{k}-a_{\ell}\right)z^{b_{k}}}{\left(\prod\limits_{% \ell=m}^{q-1}\Gamma\left(1+b_{k}-b_{\ell+1}\right)\prod\limits_{\ell=n}^{p-1}% \Gamma\left(a_{\ell+1}-b_{k}\right)\right)}.}}
    0 references
    DLMF id
    DLMF:16.17.E3
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1bdec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1bdec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1bdec
    0 references
    Q11693
    Defining formula
    a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
    xml-id
    C16.S1.XMD4.m1bdec
    0 references
    Q11694
    Defining formula
    b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
    xml-id
    C16.S1.XMD5.m1bdec
    0 references
     
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