DLMF:13.2.E2 (Q4989)

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DLMF:13.2.E2
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    Statements

    DLMF defining formula
    M ⁡ ( a , b , z ) = ∑ s = 0 ∞ ( a ) s ( b ) s ⁢ s ! ⁢ z s = 1 + a b ⁢ z + a ⁢ ( a + 1 ) b ⁢ ( b + 1 ) ⁢ 2 ! ⁢ z 2 + ⋯ , Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 𝑠 superscript 𝑧 𝑠 1 𝑎 𝑏 𝑧 𝑎 𝑎 1 𝑏 𝑏 1 2 superscript 𝑧 2 ⋯ {\displaystyle{\displaystyle M\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}}{{\left(b\right)_{s}}s!}z^{s}=1+\frac{a}{b}z+\frac{a(a+1)}% {b(b+1)2!}z^{2}+\cdots,}}
    0 references
    Symbols used
    = M ( a , b , z ) notation for the Kummer confluent hypergeometric function
    DLMF defining formula
    F 1 1 ⁡ ( a ; b ; z ) Kummer-confluent-hypergeometric-M-as-1F1 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}% \right)}}
    xml-id
    C16.S2.m7adec
    0 references
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    nonnegative integer
    DLMF defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1dec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1adec
    0 references

    Identifiers

     
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