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DLMF:13.2.E31 - MaRDI portal

DLMF:13.2.E31 (Q5020)

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DLMF:13.2.E31
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    Statements

    DLMF 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 ( a + n + 1 ) k ( n + 2 ) k ⁢ k ! ⁢ z n + k + 1 ⁢ ( ln ⁡ z + ψ ⁡ ( - a - n - k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k + 2 ) ) + ( - 1 ) n - a ⁢ ( - a - n - 1 ) ! ⁢ ∑ k = - a - n ∞ ( k + a + n ) ! ( n + 2 ) k ⁢ k ! ⁢ z n + k + 1 , 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 𝑎 𝑛 1 Pochhammer 𝑎 𝑛 1 𝑘 Pochhammer 𝑛 2 𝑘 𝑘 superscript 𝑧 𝑛 𝑘 1 𝑧 digamma 𝑎 𝑛 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 2 superscript 1 𝑛 𝑎 𝑎 𝑛 1 superscript subscript 𝑘 𝑎 𝑛 𝑘 𝑎 𝑛 Pochhammer 𝑛 2 𝑘 𝑘 superscript 𝑧 𝑛 𝑘 1 {\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}^{-a-n-1}\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\right)-\psi% \left(1+k\right)-\psi\left(n+k+2\right)\right)+(-1)^{n-a}{(-a-n-1)!}\sum_{k=-a% -n}^{\infty}\frac{(k+a+n)!}{{\left(n+2\right)_{k}}k!}z^{n+k+1},}}
    0 references
    Symbols used
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aldec
    0 references
    psi (or digamma) function
    DLMF defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aedec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5ahdec
    0 references
    principal branch of logarithm function
    DLMF defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2agdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1ldec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1addec
    0 references
     
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