DLMF:13.7.E1 (Q4409)

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DLMF:13.7.E1
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    Statements

    Defining formula
    𝐌 ⁑ ( a , b , x ) ∼ e x ⁒ x a - b Ξ“ ⁑ ( a ) ⁒ βˆ‘ s = 0 ∞ ( 1 - a ) s ⁒ ( b - a ) s s ! ⁒ x - s , asymptotic-expansion Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 π‘₯ superscript 𝑒 π‘₯ superscript π‘₯ π‘Ž 𝑏 Euler-Gamma π‘Ž superscript subscript 𝑠 0 Pochhammer 1 π‘Ž 𝑠 Pochhammer 𝑏 π‘Ž 𝑠 𝑠 superscript π‘₯ 𝑠 {\displaystyle{\displaystyle{\mathbf{M}}\left(a,b,x\right)\sim\frac{e^{x}x^{a-% b}}{\Gamma\left(a\right)}\sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left% (b-a\right)_{s}}}{s!}x^{-s},}}
    0 references
    DLMF id
    DLMF:13.7.E1
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁑ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M π‘Ž 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer π‘Ž 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11558
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C13.S1.XMD3.m1dec
    0 references
    Q11566
    Defining formula
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1dec
    0 references
     
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