DLMF:15.10.E9 (Q5125): Difference between revisions

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Property / constraint
 

b ≠ n - 1 , n - 2 , … , 0 , - 1 , - 2 , … 𝑏 𝑛 1 𝑛 2 … 0 1 2 … {\displaystyle{\displaystyle b\neq n-1,n-2,\dots,0,-1,-2,\dots}}

b\neq n-1,n-2,\dots,0,-1,-2,\dots
Property / constraint: b ≠ n - 1 , n - 2 , … , 0 , - 1 , - 2 , … 𝑏 𝑛 1 𝑛 2 … 0 1 2 … {\displaystyle{\displaystyle b\neq n-1,n-2,\dots,0,-1,-2,\dots}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
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)}}

\hyperF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
xml-id: C15.S2.E1.m2addec
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.m1aadec
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.m2aadec
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.m5aadec
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.m2aadec
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.m1adec
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.m1hdec
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.m1hdec
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.m1hdec
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.m1adec
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.m1adec

Latest revision as of 13:14, 2 January 2020

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Language Label Description Also known as
English
DLMF:15.10.E9
No description defined

    Statements

    test
    F ⁡ ( - m , b n ; z ) ⁢ ln ⁡ z - ∑ k = 1 n - 1 ( n - 1 ) ! ⁢ ( k - 1 ) ! ( n - k - 1 ) ! ⁢ ( m + 1 ) k ⁢ ( 1 - b ) k ⁢ ( - z ) - k + ∑ k = 0 m ( - m ) k ⁢ ( b ) k ( n ) k ⁢ k ! ⁢ z k ⁢ ( ψ ⁡ ( 1 + m - k ) + ψ ⁡ ( b + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k ) ) + ( - 1 ) m ⁢ m ! ⁢ ∑ k = m + 1 ∞ ( k - 1 - m ) ! ⁢ ( b ) k ( n ) k ⁢ k ! ⁢ z k , Gauss-hypergeometric-F 𝑚 𝑏 𝑛 𝑧 𝑧 superscript subscript 𝑘 1 𝑛 1 𝑛 1 𝑘 1 𝑛 𝑘 1 Pochhammer 𝑚 1 𝑘 Pochhammer 1 𝑏 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 𝑚 Pochhammer 𝑚 𝑘 Pochhammer 𝑏 𝑘 Pochhammer 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 digamma 1 𝑚 𝑘 digamma 𝑏 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 superscript 1 𝑚 𝑚 superscript subscript 𝑘 𝑚 1 𝑘 1 𝑚 Pochhammer 𝑏 𝑘 Pochhammer 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 {\displaystyle{\displaystyle F\left({-m,b\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(1-b\right)_{k}}}(-z% )^{-k}+\sum_{k=0}^{m}\frac{{\left(-m\right)_{k}}{\left(b\right)_{k}}}{{\left(n% \right)_{k}}k!}z^{k}\left(\psi\left(1+m-k\right)+\psi\left(b+k\right)-\psi% \left(1+k\right)-\psi\left(n+k\right)\right)+(-1)^{m}m!\sum_{k=m+1}^{\infty}% \frac{(k-1-m)!{\left(b\right)_{k}}}{{\left(n\right)_{k}}k!}z^{k},}}
    0 references
    DLMF id
    DLMF:15.10.E9
    0 references
    constraint
    b ≠ n - 1 , n - 2 , … , 0 , - 1 , - 2 , … 𝑏 𝑛 1 𝑛 2 … 0 1 2 … {\displaystyle{\displaystyle b\neq n-1,n-2,\dots,0,-1,-2,\dots}}
    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.m2addec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11650
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1adec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1hdec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1hdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1hdec
    0 references
    Q11658
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1adec
    0 references
    Q11651
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1adec
    0 references
     
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