DLMF:15.2.E4 (Q4981): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
Defining formula:

( 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: Q10754 / rank
 
Normal rank
Property / Symbols used: Q10754 / qualifier
 
Defining formula:

( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}

\binom{\NVar{m}}{\NVar{n}}
Property / Symbols used: Q10754 / qualifier
 
xml-id: C1.S2.SS1.m1adec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5abdec
Property / Symbols used
 
Property / Symbols used: Q11650 / rank
 
Normal rank
Property / Symbols used: Q11650 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11650 / qualifier
 
xml-id: C15.S1.XMD10.m1dec
Property / Symbols used
 
Property / Symbols used: Q11644 / rank
 
Normal rank
Property / Symbols used: Q11644 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11644 / qualifier
 
xml-id: C15.S1.XMD3.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11646 / rank
 
Normal rank
Property / Symbols used: Q11646 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
 
xml-id: C15.S1.XMD5.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11647 / rank
 
Normal rank
Property / Symbols used: Q11647 / qualifier
 
Defining formula:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q11647 / qualifier
 
xml-id: C15.S1.XMD6.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11651 / rank
 
Normal rank
Property / Symbols used: Q11651 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11651 / qualifier
 
xml-id: C15.S1.XMD9.m1dec

Latest revision as of 13:46, 2 January 2020

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DLMF:15.2.E4
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    Statements

    F ( - m , b ; c ; z ) = n = 0 m ( - m ) n ( b ) n ( c ) n n ! z n = n = 0 m ( - 1 ) n ( m n ) ( b ) n ( c ) n z n . Gauss-hypergeometric-F 𝑚 𝑏 𝑐 𝑧 superscript subscript 𝑛 0 𝑚 Pochhammer 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑛 superscript subscript 𝑛 0 𝑚 superscript 1 𝑛 binomial 𝑚 𝑛 Pochhammer 𝑏 𝑛 Pochhammer 𝑐 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle F\left(-m,b;c;z\right)=\sum_{n=0}^{m}\frac{{\left% (-m\right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}{n!}}z^{n}=\sum_{n=0}% ^{m}(-1)^{n}\genfrac{(}{)}{0.0pt}{}{m}{n}\frac{{\left(b\right)_{n}}}{{\left(c% \right)_{n}}}z^{n}.}}
    0 references
    DLMF:15.2.E4
    0 references
    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)}}
    C15.S2.E1.m2aadec
    0 references
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    C5.S2.SS3.m1abdec
    0 references
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    C1.S2.SS1.m1adec
    0 references
    ! {\displaystyle{\displaystyle!}}
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C15.S1.XMD10.m1dec
    0 references
    z 𝑧 {\displaystyle{\displaystyle z}}
    C15.S1.XMD3.m1bdec
    0 references
    b 𝑏 {\displaystyle{\displaystyle b}}
    C15.S1.XMD5.m1cdec
    0 references
    c 𝑐 {\displaystyle{\displaystyle c}}
    C15.S1.XMD6.m1cdec
    0 references
    m 𝑚 {\displaystyle{\displaystyle m}}
    C15.S1.XMD9.m1dec
    0 references