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

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Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
test:

( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}

\Pochhammersym{\NVar{a}}{\NVar{n}}
Property / Symbols used: Q10759 / qualifier
 
xml-id: C5.S2.SS3.m1abdec
Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

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.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5abdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11650 / rank
 
Normal rank
Property / Symbols used: Q11650 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11650 / qualifier
 
xml-id: C15.S1.XMD10.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11644 / rank
 
Normal rank
Property / Symbols used: Q11644 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11644 / qualifier
 
xml-id: C15.S1.XMD3.m1idec
Property / Symbols used
 
Property / Symbols used: Q11645 / rank
 
Normal rank
Property / Symbols used: Q11645 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11645 / qualifier
 
xml-id: C15.S1.XMD4.m1idec
Property / Symbols used
 
Property / Symbols used: Q11646 / rank
 
Normal rank
Property / Symbols used: Q11646 / qualifier
 
test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
 
xml-id: C15.S1.XMD5.m1idec
Property / Symbols used
 
Property / Symbols used: Q11658 / rank
 
Normal rank
Property / Symbols used: Q11658 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11658 / qualifier
 
xml-id: C15.S1.XMD7.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11652 / rank
 
Normal rank
Property / Symbols used: Q11652 / qualifier
 
test:

ℓ ℓ {\displaystyle{\displaystyle\ell}}

\ell
Property / Symbols used: Q11652 / qualifier
 
xml-id: C15.S1.XMD8.m1dec
Property / Symbols used
 
Property / Symbols used: Q11651 / rank
 
Normal rank
Property / Symbols used: Q11651 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11651 / qualifier
 
xml-id: C15.S1.XMD9.m1bdec

Latest revision as of 13:14, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:15.10.E10
No description defined

    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
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2aedec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q11650
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1bdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1idec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1idec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1idec
    0 references
    Q11658
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1bdec
    0 references
    Q11652
    test
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C15.S1.XMD8.m1dec
    0 references
    Q11651
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1bdec
    0 references
     
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