DLMF:13.13.E3 (Q4480)

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DLMF:13.13.E3
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    Statements

    test
    ( x x + y ) a ⁢ ∑ n = 0 ∞ ( a ) n ⁢ y n n ! ⁢ ( x + y ) n ⁢ M ⁡ ( a + n , b , x ) , superscript 𝑥 𝑥 𝑦 𝑎 superscript subscript 𝑛 0 Pochhammer 𝑎 𝑛 superscript 𝑦 𝑛 𝑛 superscript 𝑥 𝑦 𝑛 Kummer-confluent-hypergeometric-M 𝑎 𝑛 𝑏 𝑥 {\displaystyle{\displaystyle\left(\frac{x}{x+y}\right)^{a}\sum_{n=0}^{\infty}% \frac{{\left(a\right)_{n}}y^{n}}{n!(x+y)^{n}}M\left(a+n,b,x\right),}}
    0 references
    DLMF id
    DLMF:13.13.E3
    0 references
    constraint
    ℜ ⁡ ( y / x ) > - 1 2 𝑦 𝑥 1 2 {\displaystyle{\displaystyle\Re(y/x)>-\tfrac{1}{2}}}
    0 references
    Symbols used
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ 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.m2abdec
    0 references
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
     
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