DLMF:15.2.E1 (Q4977): Difference between revisions

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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.m5adec
Property / Symbols used
 
Property / Symbols used: Q11648 / rank
 
Normal rank
Property / Symbols used: Q11648 / qualifier
 
test:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11648 / qualifier
 
xml-id: C15.S1.XMD11.m1dec
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.m1dec
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.m1dec
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.m1dec
Property / Symbols used
 
Property / Symbols used: Q11647 / rank
 
Normal rank
Property / Symbols used: Q11647 / qualifier
 
test:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q11647 / qualifier
 
xml-id: C15.S1.XMD6.m1dec

Latest revision as of 12:45, 2 January 2020

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Language Label Description Also known as
English
DLMF:15.2.E1
No description defined

    Statements

    test
    F ⁡ ( a , b ; c ; z ) = ∑ s = 0 ∞ ( a ) s ⁢ ( b ) s ( c ) s ⁢ s ! ⁢ z s = 1 + a ⁢ b c ⁢ z + a ⁢ ( a + 1 ) ⁢ b ⁢ ( b + 1 ) c ⁢ ( c + 1 ) ⁢ 2 ! ⁢ z 2 + ⋯ = Γ ⁡ ( c ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ ∑ s = 0 ∞ Γ ⁡ ( a + s ) ⁢ Γ ⁡ ( b + s ) Γ ⁡ ( c + s ) ⁢ s ! ⁢ z s , Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 Pochhammer 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 1 𝑎 𝑏 𝑐 𝑧 𝑎 𝑎 1 𝑏 𝑏 1 𝑐 𝑐 1 2 superscript 𝑧 2 ⋯ Euler-Gamma 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑏 superscript subscript 𝑠 0 Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}{\left(b\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{% ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{% \Gamma\left(a\right)\Gamma\left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(% a+s\right)\Gamma\left(b+s\right)}{\Gamma\left(c+s\right)s!}z^{s},}}
    0 references
    DLMF id
    DLMF:15.2.E1
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q11581
    test
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5adec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11648
    test
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C15.S1.XMD11.m1dec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1dec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1dec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1dec
    0 references
    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1dec
    0 references
     
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