DLMF:13.2.E30 (Q4322)

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DLMF:13.2.E30
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    Statements

    Defining formula
    ∑ k = 1 n + 1 ( n + 1 ) ! ⁢ ( k - 1 ) ! ( n - k + 1 ) ! ⁢ ( - a - n ) k ⁢ z n - k + 1 - ∑ k = 0 ∞ ( a + n + 1 ) k ( n + 2 ) k ⁢ k ! ⁢ z n + k + 1 ⁢ ( ln ⁡ z + ψ ⁡ ( a + n + k + 1 ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k + 2 ) ) , superscript subscript 𝑘 1 𝑛 1 𝑛 1 𝑘 1 𝑛 𝑘 1 Pochhammer 𝑎 𝑛 𝑘 superscript 𝑧 𝑛 𝑘 1 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑛 1 𝑘 Pochhammer 𝑛 2 𝑘 𝑘 superscript 𝑧 𝑛 𝑘 1 𝑧 digamma 𝑎 𝑛 𝑘 1 digamma 1 𝑘 digamma 𝑛 𝑘 2 {\displaystyle{\displaystyle\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left% (-a-n\right)_{k}}}z^{n-k+1}-\sum_{k=0}^{\infty}\frac{{\left(a+n+1\right)_{k}}}% {{\left(n+2\right)_{k}}k!}z^{n+k+1}\left(\ln z+\psi\left(a+n+k+1\right)-\psi% \left(1+k\right)-\psi\left(n+k+2\right)\right),}}
    0 references
    DLMF id
    DLMF:13.2.E30
    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.m1akdec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2addec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5agdec
    0 references
     
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