DLMF:15.10.E12 (Q5128): 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.m1adec

Latest revision as of 14:14, 2 January 2020

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DLMF:15.10.E12
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    Statements

    Defining formula
    w 2 ⁢ ( z ) = z 1 - c ⁢ F ⁡ ( a - c + 1 , b - c + 1 2 - c ; z ) = z 1 - c ⁢ ( 1 - z ) c - a - b ⁢ F ⁡ ( 1 - a , 1 - b 2 - c ; z ) = z 1 - c ⁢ ( 1 - z ) c - a - 1 ⁢ F ⁡ ( a - c + 1 , 1 - b 2 - c ; z z - 1 ) = z 1 - c ⁢ ( 1 - z ) c - b - 1 ⁢ F ⁡ ( 1 - a , b - c + 1 2 - c ; z z - 1 ) . subscript 𝑤 2 𝑧 superscript 𝑧 1 𝑐 Gauss-hypergeometric-F 𝑎 𝑐 1 𝑏 𝑐 1 2 𝑐 𝑧 superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑎 𝑏 Gauss-hypergeometric-F 1 𝑎 1 𝑏 2 𝑐 𝑧 superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑎 1 Gauss-hypergeometric-F 𝑎 𝑐 1 1 𝑏 2 𝑐 𝑧 𝑧 1 superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑏 1 Gauss-hypergeometric-F 1 𝑎 𝑏 𝑐 1 2 𝑐 𝑧 𝑧 1 {\displaystyle{\displaystyle w_{2}(z)={z^{1-c}}F\left({a-c+1,b-c+1\atop 2-c};z% \right)={z^{1-c}(1-z)^{c-a-b}}\*F\left({1-a,1-b\atop 2-c};z\right)={z^{1-c}(1-% z)^{c-a-1}}\*F\left({a-c+1,1-b\atop 2-c};\frac{z}{z-1}\right)={z^{1-c}(1-z)^{c% -b-1}}\*F\left({1-a,b-c+1\atop 2-c};\frac{z}{z-1}\right).}}
    0 references
    DLMF id
    DLMF:15.10.E12
    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.m2agdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1kdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1kdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1kdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1hdec
    0 references
    Q11667
    Defining formula
    w m ⁢ ( z ) subscript 𝑤 𝑚 𝑧 {\displaystyle{\displaystyle w_{m}(z)}}
    xml-id
    C15.S10.XMD2.m1adec
    0 references
     
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