DLMF:15.10.E10 (Q5126): Difference between revisions

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Property / constraint
 

ℓ = 0 , 1 , 2 , … , m ℓ 0 1 2 … 𝑚 {\displaystyle{\displaystyle\ell=0,1,2,\dots,m}}

\ell=0,1,2,\dots,m
Property / constraint: ℓ = 0 , 1 , 2 , … , m ℓ 0 1 2 … 𝑚 {\displaystyle{\displaystyle\ell=0,1,2,\dots,m}} / rank
 
Normal rank

Revision as of 16:57, 30 December 2019

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DLMF:15.10.E10
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    Statements

    test
    F ⁡ ( - m , - ℓ n ; z ) ⁢ ln ⁡ z - ∑ k = 1 n - 1 ( n - 1 ) ! ⁢ ( k - 1 ) ! ( n - k - 1 ) ! ⁢ ( m + 1 ) k ⁢ ( ℓ + 1 ) k ⁢ ( - z ) - k + ∑ k = 0 ℓ ( - m ) k ⁢ ( - ℓ ) k ( n ) k ⁢ k ! ⁢ z k ⁢ ( ψ ⁡ ( 1 + m - k ) + ψ ⁡ ( 1 + ℓ - k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k ) ) + ( - 1 ) ℓ ⁢ ℓ ! ⁢ ∑ k = ℓ + 1 m ( k - 1 - ℓ ) ! ⁢ ( - m ) k ( n ) k ⁢ k ! ⁢ z k , Gauss-hypergeometric-F 𝑚 ℓ 𝑛 𝑧 𝑧 superscript subscript 𝑘 1 𝑛 1 𝑛 1 𝑘 1 𝑛 𝑘 1 Pochhammer 𝑚 1 𝑘 Pochhammer ℓ 1 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 ℓ Pochhammer 𝑚 𝑘 Pochhammer ℓ 𝑘 Pochhammer 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 digamma 1 𝑚 𝑘 digamma 1 ℓ 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 superscript 1 ℓ ℓ superscript subscript 𝑘 ℓ 1 𝑚 𝑘 1 ℓ Pochhammer 𝑚 𝑘 Pochhammer 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle F\left({-m,-\ell\atop n};z\right)\ln z-\sum_{k=1}% ^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(m+1\right)_{k}}{\left(\ell+1\right)_{% k}}}(-z)^{-k}+\sum_{k=0}^{\ell}\frac{{\left(-m\right)_{k}}{\left(-\ell\right)_% {k}}}{{\left(n\right)_{k}}k!}z^{k}\left(\psi\left(1+m-k\right)+\psi\left(1+% \ell-k\right)-\psi\left(1+k\right)-\psi\left(n+k\right)\right)+(-1)^{\ell}\ell% \,!\sum_{k=\ell+1}^{m}\frac{(k-1-\ell)!{\left(-m\right)_{k}}}{{\left(n\right)_% {k}}k!}z^{k},}}
    0 references
    DLMF id
    DLMF:15.10.E10
    0 references
    constraint
    ℓ = 0 , 1 , 2 , … , m ℓ 0 1 2 … 𝑚 {\displaystyle{\displaystyle\ell=0,1,2,\dots,m}}
    0 references
     
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