DLMF:13.2.E5 (Q4295)

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DLMF:13.2.E5
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    Statements

    Defining formula
    lim b → - n ⁡ M ⁡ ( a , b , z ) Γ ⁡ ( b ) = 𝐌 ⁡ ( a , - n , z ) = ( a ) n + 1 ( n + 1 ) ! ⁢ z n + 1 ⁢ M ⁡ ( a + n + 1 , n + 2 , z ) . subscript → 𝑏 𝑛 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 Euler-Gamma 𝑏 Kummer-confluent-hypergeometric-bold-M 𝑎 𝑛 𝑧 Pochhammer 𝑎 𝑛 1 𝑛 1 superscript 𝑧 𝑛 1 Kummer-confluent-hypergeometric-M 𝑎 𝑛 1 𝑛 2 𝑧 {\displaystyle{\displaystyle\lim_{b\to-n}\frac{M\left(a,b,z\right)}{\Gamma% \left(b\right)}={\mathbf{M}}\left(a,-n,z\right)=\frac{{\left(a\right)_{n+1}}}{% (n+1)!}z^{n+1}M\left(a+n+1,n+2,z\right).}}
    0 references
    DLMF id
    DLMF:13.2.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    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
    Olver’s confluent hypergeometric function
    Defining formula
    𝐌 ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-bold-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right% )}}
    xml-id
    C13.S2.E3.m2abdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1dec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1ddec
    0 references
     
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