DLMF:35.7.E3 (Q9814)

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DLMF:35.7.E3
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    Statements

    Defining formula
    F 1 2 ⁡ ( a , b c ; [ t 1 0 0 t 2 ] ) = ∑ k = 0 ∞ ( a ) k ⁢ ( c - a ) k ⁢ ( b ) k ⁢ ( c - b ) k k ! ⁢ ( c ) 2 ⁢ k ⁢ ( c - 1 2 ) k ⁢ ( t 1 ⁢ t 2 ) k ⁢ F 1 2 ⁡ ( a + k , b + k c + 2 ⁢ k ; t 1 + t 2 - t 1 ⁢ t 2 ) . Gauss-hypergeometric-of-matrix-pFq 2 1 𝑎 𝑏 𝑐 subscript 𝑡 1 0 0 subscript 𝑡 2 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑘 Pochhammer 𝑐 𝑎 𝑘 Pochhammer 𝑏 𝑘 Pochhammer 𝑐 𝑏 𝑘 𝑘 Pochhammer 𝑐 2 𝑘 Pochhammer 𝑐 1 2 𝑘 superscript subscript 𝑡 1 subscript 𝑡 2 𝑘 Gauss-hypergeometric-of-matrix-pFq 2 1 𝑎 𝑘 𝑏 𝑘 𝑐 2 𝑘 subscript 𝑡 1 subscript 𝑡 2 subscript 𝑡 1 subscript 𝑡 2 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{% 1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).}}
    0 references
    DLMF id
    DLMF:35.7.E3
    0 references
    Symbols used
    generalized hypergeometric function of matrix argument
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; 𝐓 ) Gauss-hypergeometric-of-matrix-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝐓 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}
    xml-id
    C35.S8.E1.m2abdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    Q12726
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1adec
    0 references
    Q12737
    Defining formula
    t j subscript 𝑡 𝑗 {\displaystyle{\displaystyle t_{j}}}
    xml-id
    C35.S1.XMD14.m1dec
    0 references
    Q12728
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1adec
    0 references
    Q12736
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C35.S1.XMD3.m1adec
    0 references
    Q12729
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C35.S1.XMD5.m1adec
    0 references
     
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