DLMF:15.10.E8 (Q5124)

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DLMF:15.10.E8
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    Statements

    test
    F ⁡ ( a , b n ; z ) ⁢ ln ⁡ z - ∑ k = 1 n - 1 ( n - 1 ) ! ⁢ ( k - 1 ) ! ( n - k - 1 ) ! ⁢ ( 1 - a ) k ⁢ ( 1 - b ) k ⁢ ( - z ) - k + ∑ k = 0 ∞ ( a ) k ⁢ ( b ) k ( n ) k ⁢ k ! ⁢ z k ⁢ ( ψ ⁡ ( a + k ) + ψ ⁡ ( b + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k ) ) , Gauss-hypergeometric-F 𝑎 𝑏 𝑛 𝑧 𝑧 superscript subscript 𝑘 1 𝑛 1 𝑛 1 𝑘 1 𝑛 𝑘 1 Pochhammer 1 𝑎 𝑘 Pochhammer 1 𝑏 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑘 Pochhammer 𝑏 𝑘 Pochhammer 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 digamma 𝑎 𝑘 digamma 𝑏 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 {\displaystyle{\displaystyle F\left({a,b\atop n};z\right)\ln z-\sum_{k=1}^{n-1% }\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(1-a\right)_{k}}{\left(1-b\right)_{k}}}(-z)% ^{-k}\\ +\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b\right)_{k}}}{{\left(n% \right)_{k}}k!}z^{k}\left(\psi\left(a+k\right)+\psi\left(b+k\right)-\psi\left(% 1+k\right)-\psi\left(n+k\right)\right),}}
    0 references
    DLMF id
    DLMF:15.10.E8
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    constraint
    a , b ≠ n - 1 , n - 2 , … , 0 , - 1 , - 2 , … formulae-sequence 𝑎 𝑏 𝑛 1 𝑛 2 … 0 1 2 … {\displaystyle{\displaystyle a,b\neq n-1,n-2,\dots,0,-1,-2,\dots}}
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    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2acdec
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    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
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    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q11650
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1dec
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    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1gdec
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    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1gdec
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    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1gdec
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    Q11658
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1dec
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    Q11651
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1dec
    0 references
     
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