DLMF:25.11.E32 (Q7706)

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DLMF:25.11.E32
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    Statements

    Defining formula
    ∫ 0 a x n ⁢ ψ ⁡ ( x ) ⁢ d x = ( - 1 ) n - 1 ⁢ ζ ′ ⁡ ( - n ) + ( - 1 ) n ⁢ h ⁢ ( n ) ⁢ B n + 1 n + 1 - ∑ k = 0 n ( - 1 ) k ⁢ ( n k ) ⁢ h ⁢ ( k ) ⁢ B k + 1 ⁢ ( a ) k + 1 ⁢ a n - k + ∑ k = 0 n ( - 1 ) k ⁢ ( n k ) ⁢ ζ ′ ⁡ ( - k , a ) ⁢ a n - k , superscript subscript 0 𝑎 superscript 𝑥 𝑛 digamma 𝑥 𝑥 superscript 1 𝑛 1 diffop Riemann-zeta 1 𝑛 superscript 1 𝑛 ℎ 𝑛 Bernoulli-number-B 𝑛 1 𝑛 1 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 binomial 𝑛 𝑘 ℎ 𝑘 Bernoulli-number-B 𝑘 1 𝑎 𝑘 1 superscript 𝑎 𝑛 𝑘 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 binomial 𝑛 𝑘 diffop Hurwitz-zeta 1 𝑘 𝑎 superscript 𝑎 𝑛 𝑘 {\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1% )^{n-1}\zeta'\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1% )^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}% ^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},}}
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    DLMF id
    DLMF:25.11.E32
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\dots}}
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    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1ahdec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aagdec
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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2agdec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
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    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2acdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1apdec
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    Q12150
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1ddec
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