DLMF:15.10.E13 (Q5129): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11667 / rank
 
Normal rank
Property / Symbols used: Q11667 / qualifier
 
Defining formula:

w m ⁢ ( z ) subscript 𝑤 𝑚 𝑧 {\displaystyle{\displaystyle w_{m}(z)}}

w_{m}(z)
Property / Symbols used: Q11667 / qualifier
 
xml-id: C15.S10.XMD2.m1bdec

Latest revision as of 14:15, 2 January 2020

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DLMF:15.10.E13
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    Statements

    Defining formula
    w 3 ⁢ ( z ) = F ⁡ ( a , b a + b - c + 1 ; 1 - z ) = z 1 - c ⁢ F ⁡ ( a - c + 1 , b - c + 1 a + b - c + 1 ; 1 - z ) = z - a ⁢ F ⁡ ( a , a - c + 1 a + b - c + 1 ; 1 - 1 z ) = z - b ⁢ F ⁡ ( b , b - c + 1 a + b - c + 1 ; 1 - 1 z ) . subscript 𝑤 3 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑎 𝑏 𝑐 1 1 𝑧 superscript 𝑧 1 𝑐 Gauss-hypergeometric-F 𝑎 𝑐 1 𝑏 𝑐 1 𝑎 𝑏 𝑐 1 1 𝑧 superscript 𝑧 𝑎 Gauss-hypergeometric-F 𝑎 𝑎 𝑐 1 𝑎 𝑏 𝑐 1 1 1 𝑧 superscript 𝑧 𝑏 Gauss-hypergeometric-F 𝑏 𝑏 𝑐 1 𝑎 𝑏 𝑐 1 1 1 𝑧 {\displaystyle{\displaystyle w_{3}(z)=F\left({a,b\atop a+b-c+1};1-z\right)=z^{% 1-c}F\left({a-c+1,b-c+1\atop a+b-c+1};1-z\right)=z^{-a}F\left({a,a-c+1\atop a+% b-c+1};1-\frac{1}{z}\right)=z^{-b}F\left({b,b-c+1\atop a+b-c+1};1-\frac{1}{z}% \right).}}
    0 references
    DLMF id
    DLMF:15.10.E13
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    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.m2ahdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ldec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1ldec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1ldec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1idec
    0 references
    Q11667
    Defining formula
    w m ⁢ ( z ) subscript 𝑤 𝑚 𝑧 {\displaystyle{\displaystyle w_{m}(z)}}
    xml-id
    C15.S10.XMD2.m1bdec
    0 references
     
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