DLMF:25.11.E38 (Q7712): Difference between revisions

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Property / constraint
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| z | < | a | 𝑧 π‘Ž {\displaystyle{\displaystyle|z|<|a|}}

|z|<|a|
Property / constraint: | z | < | a | 𝑧 π‘Ž {\displaystyle{\displaystyle|z|<|a|}} / rank
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Normal rank
Property / Symbols used
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Property / Symbols used: Hurwitz zeta function / rank
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Normal rank
Property / Symbols used: Hurwitz zeta function / qualifier
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test:

ΞΆ ⁑ ( s , a ) Hurwitz-zeta 𝑠 π‘Ž {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}

\Hurwitzzeta@{\NVar{s}}{\NVar{a}}
Property / Symbols used: Hurwitz zeta function / qualifier
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xml-id: C25.S11.E1.m2aaldec
Property / Symbols used
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Property / Symbols used: Q10754 / rank
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Normal rank
Property / Symbols used: Q10754 / qualifier
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test:

( m n ) binomial π‘š 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}

\binom{\NVar{m}}{\NVar{n}}
Property / Symbols used: Q10754 / qualifier
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xml-id: C1.S2.SS1.m1abdec
Property / Symbols used
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Property / Symbols used: psi (or digamma) function / rank
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Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
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test:

ψ ⁑ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
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xml-id: C5.S2.E2.m2addec
Property / Symbols used
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Property / Symbols used: Q10755 / rank
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Normal rank
Property / Symbols used: Q10755 / qualifier
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test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
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xml-id: introduction.Sx4.p1.t1.r15.m5aedec
Property / Symbols used
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Property / Symbols used: Q10811 / rank
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Normal rank
Property / Symbols used: Q10811 / qualifier
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test:

β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
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xml-id: C1.S9.E2.m1audec
Property / Symbols used
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Property / Symbols used: Q12150 / rank
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Normal rank
Property / Symbols used: Q12150 / qualifier
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test:

k π‘˜ {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12150 / qualifier
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xml-id: C25.S1.XMD1.m1hdec
Property / Symbols used
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Property / Symbols used: Q12148 / rank
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Normal rank
Property / Symbols used: Q12148 / qualifier
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test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12148 / qualifier
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xml-id: C25.S1.XMD3.m1udec
Property / Symbols used
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Property / Symbols used: Q12161 / rank
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Normal rank
Property / Symbols used: Q12161 / qualifier
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test:

a π‘Ž {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12161 / qualifier
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xml-id: C25.S1.XMD6.m1aedec
Property / Symbols used
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Property / Symbols used: Q12156 / rank
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Normal rank
Property / Symbols used: Q12156 / qualifier
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test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12156 / qualifier
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xml-id: C25.S1.XMD8.m1bdec

Latest revision as of 12:46, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:25.11.E38
No description defined

    Statements

    test
    βˆ‘ k = 1 ∞ ( n + k k ) ⁒ ΞΆ ⁑ ( n + k + 1 , a ) ⁒ z k = ( - 1 ) n n ! ⁒ ( ψ ( n ) ⁑ ( a ) - ψ ( n ) ⁑ ( a - z ) ) , superscript subscript π‘˜ 1 binomial 𝑛 π‘˜ π‘˜ Hurwitz-zeta 𝑛 π‘˜ 1 π‘Ž superscript 𝑧 π‘˜ superscript 1 𝑛 𝑛 digamma 𝑛 π‘Ž digamma 𝑛 π‘Ž 𝑧 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}% \zeta\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a% \right)-{\psi^{(n)}}\left(a-z\right)\right),}}
    0 references
    DLMF id
    DLMF:25.11.E38
    0 references
    constraint
    β„œ ⁑ a > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    β„œ ⁑ a > 0 π‘Ž 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    | z | < | a | 𝑧 π‘Ž {\displaystyle{\displaystyle|z|<|a|}}
    0 references
    Symbols used
    Hurwitz zeta function
    test
    ΞΆ ⁑ ( s , a ) Hurwitz-zeta 𝑠 π‘Ž {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aaldec
    0 references
    Q10754
    test
    ( m n ) binomial π‘š 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1abdec
    0 references
    psi (or digamma) function
    test
    ψ ⁑ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2addec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aedec
    0 references
    Q10811
    test
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1audec
    0 references
    Q12150
    test
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C25.S1.XMD1.m1hdec
    0 references
    Q12148
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1udec
    0 references
    Q12161
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1aedec
    0 references
    Q12156
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C25.S1.XMD8.m1bdec
    0 references
     
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