DLMF:13.2.E9 (Q4299)

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DLMF:13.2.E9
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    Statements

    test
    U ⁡ ( a , n + 1 , z ) = ( - 1 ) n + 1 n ! ⁢ Γ ⁡ ( a - n ) ⁢ ∑ k = 0 ∞ ( a ) k ( n + 1 ) k ⁢ k ! ⁢ z k ⁢ ( ln ⁡ z + ψ ⁡ ( a + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k + 1 ) ) + 1 Γ ⁡ ( a ) ⁢ ∑ k = 1 n ( k - 1 ) ! ⁢ ( 1 - a + k ) n - k ( n - k ) ! ⁢ z - k . Kummer-confluent-hypergeometric-U 𝑎 𝑛 1 𝑧 superscript 1 𝑛 1 𝑛 Euler-Gamma 𝑎 𝑛 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 1 Euler-Gamma 𝑎 superscript subscript 𝑘 1 𝑛 𝑘 1 Pochhammer 1 𝑎 𝑘 𝑛 𝑘 𝑛 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma% \left(a-n\right)}\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)+\frac{1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k% -1)!{\left(1-a+k\right)_{n-k}}}{(n-k)!}z^{-k}.}}
    0 references
    DLMF id
    DLMF:13.2.E9
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Kummer confluent hypergeometric function
    test
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2acdec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aedec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
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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
    Q11559
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1cdec
    0 references
     
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