DLMF:13.2.E42 (Q4334)

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DLMF:13.2.E42
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    Statements

    test
    U ⁡ ( a , b , z ) = Γ ⁡ ( 1 - b ) Γ ⁡ ( a - b + 1 ) ⁢ M ⁡ ( a , b , z ) + Γ ⁡ ( b - 1 ) Γ ⁡ ( a ) ⁢ z 1 - b ⁢ M ⁡ ( a - b + 1 , 2 - b , z ) . Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 Euler-Gamma 1 𝑏 Euler-Gamma 𝑎 𝑏 1 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 Euler-Gamma 𝑏 1 Euler-Gamma 𝑎 superscript 𝑧 1 𝑏 Kummer-confluent-hypergeometric-M 𝑎 𝑏 1 2 𝑏 𝑧 {\displaystyle{\displaystyle U\left(a,b,z\right)=\frac{\Gamma\left(1-b\right)}% {\Gamma\left(a-b+1\right)}M\left(a,b,z\right)+\frac{\Gamma\left(b-1\right)}{% \Gamma\left(a\right)}z^{1-b}M\left(a-b+1,2-b,z\right).}}
    0 references
    DLMF id
    DLMF:13.2.E42
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ardec
    0 references
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    test
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2ajdec
    0 references
    Kummer confluent hypergeometric function
    test
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2aydec
    0 references
    Q11557
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1aodec
    0 references
     
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