DLMF:13.2.E28 (Q4320): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
Defining formula:

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

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2acdec

Revision as of 15:38, 2 January 2020

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DLMF:13.2.E28
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    Statements

    Defining formula
    ∑ k = 1 n n ! ⁢ ( k - 1 ) ! ( n - k ) ! ⁢ ( 1 - a ) k ⁢ z - k - ∑ k = 0 - a ( a ) k ( n + 1 ) k ⁢ k ! ⁢ z k ⁢ ( ln ⁡ z + ψ ⁡ ( 1 - a - k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k + 1 ) ) + ( - 1 ) 1 - a ⁢ ( - a ) ! ⁢ ∑ k = 1 - a ∞ ( k - 1 + a ) ! ( n + 1 ) k ⁢ k ! ⁢ z k , superscript subscript 𝑘 1 𝑛 𝑛 𝑘 1 𝑛 𝑘 Pochhammer 1 𝑎 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 𝑎 Pochhammer 𝑎 𝑘 Pochhammer 𝑛 1 𝑘 𝑘 superscript 𝑧 𝑘 𝑧 digamma 1 𝑎 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 1 superscript 1 1 𝑎 𝑎 superscript subscript 𝑘 1 𝑎 𝑘 1 𝑎 Pochhammer 𝑛 1 𝑘 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a% \right)_{k}}}z^{-k}-\sum_{k=0}^{-a}\frac{{\left(a\right)_{k}}}{{\left(n+1% \right)_{k}}k!}z^{k}\left(\ln z+\psi\left(1-a-k\right)-\psi\left(1+k\right)-% \psi\left(n+k+1\right)\right)+(-1)^{1-a}(-a)!\sum_{k=1-a}^{\infty}\frac{(k-1+a% )!}{{\left(n+1\right)_{k}}k!}z^{k},}}
    0 references
    DLMF id
    DLMF:13.2.E28
    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.m1ajdec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2acdec
    0 references
     
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