DLMF:13.2.E27 (Q4319)

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DLMF:13.2.E27
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    Statements

    Defining formula
    ∑ k = 1 n n ! ⁢ ( k - 1 ) ! ( n - k ) ! ⁢ ( 1 - a ) k ⁢ z - k - ∑ k = 0 ∞ ( a ) k ( n + 1 ) k ⁢ k ! ⁢ z k ⁢ ( ln ⁡ z + ψ ⁡ ( a + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k + 1 ) ) , superscript subscript 𝑘 1 𝑛 𝑛 𝑘 1 𝑛 𝑘 Pochhammer 1 𝑎 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑘 Pochhammer 𝑛 1 𝑘 𝑘 superscript 𝑧 𝑘 𝑧 digamma 𝑎 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 1 {\displaystyle{\displaystyle\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a% \right)_{k}}}z^{-k}-\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(n+1% \right)_{k}}k!}z^{k}\left(\ln z+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi% \left(n+k+1\right)\right),}}
    0 references
    DLMF id
    DLMF:13.2.E27
    0 references
    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aidec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1hdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1zdec
    0 references
     
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