DLMF:35.7.E1 (Q9812)

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DLMF:35.7.E1
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    Statements

    Defining formula
    F 1 2 ⁥ ( a , b c ; 𝐓 ) = ∑ k = 0 ∞ 1 k ! ⁹ ∑ | Îș | = k [ a ] Îș ⁹ [ b ] Îș [ c ] Îș ⁹ Z Îș ⁥ ( 𝐓 ) , Gauss-hypergeometric-of-matrix-pFq 2 1 𝑎 𝑏 𝑐 𝐓 superscript subscript 𝑘 0 1 𝑘 subscript 𝜅 𝑘 shifted-factorial 𝑎 𝜅 shifted-factorial 𝑏 𝜅 shifted-factorial 𝑐 𝜅 zonal-polynomial 𝜅 𝐓 {\displaystyle{\displaystyle{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)% =\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa% }}{\left[b\right]_{\kappa}}}{{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf% {T}\right)},}}
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    DLMF id
    DLMF:35.7.E1
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    constraint
    - c + 1 2 ⁱ ( j + 1 ) ∉ ℕ 𝑐 1 2 𝑗 1 {\displaystyle{\displaystyle-c+\frac{1}{2}(j+1)\notin\mathbb{N}}}
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    1 ≀ j ≀ m 1 𝑗 𝑚 {\displaystyle{\displaystyle 1\leq j\leq m}}
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    || 𝐓 || < 1 norm 𝐓 1 {\displaystyle{\displaystyle||\mathbf{T}||<1}}
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    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.m2adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q11245
    Defining formula
    ∉ {\displaystyle{\displaystyle\notin}}
    xml-id
    introduction.Sx4.p1.t1.r11.m2adec
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    Q12208
    Defining formula
    ℕ {\displaystyle{\displaystyle\mathbb{N}}}
    xml-id
    introduction.Sx4.p2.t1.r11.m2adec
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    partitional shifted factorial
    Defining formula
    [ a ] Îș shifted-factorial 𝑎 𝜅 {\displaystyle{\displaystyle{\left[\NVar{a}\right]_{\NVar{\kappa}}}}}
    xml-id
    C35.S4.E1.m2adec
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    Q12730
    Defining formula
    Z Îș ⁥ ( 𝐓 ) zonal-polynomial 𝜅 𝐓 {\displaystyle{\displaystyle Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)}}
    xml-id
    C35.S4.SS1.p3.m3adec
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    Q12726
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1dec
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    Q12724
    Defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1dec
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    Q12728
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1dec
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    Q12736
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C35.S1.XMD3.m1dec
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    Q12727
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C35.S1.XMD4.m1dec
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    Q12729
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C35.S1.XMD5.m1dec
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    Q12723
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1dec
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