DLMF:13.2.E8 (Q4298)

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DLMF:13.2.E8
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    Statements

    Defining formula
    U ⁡ ( a , a + n + 1 , z ) = ( - 1 ) n ⁢ ( 1 - a - n ) n z a + n ⁢ M ⁡ ( - n , 1 - a - n , z ) = z - a ⁢ ∑ s = 0 n ( n s ) ⁢ ( a ) s ⁢ z - s . Kummer-confluent-hypergeometric-U 𝑎 𝑎 𝑛 1 𝑧 superscript 1 𝑛 Pochhammer 1 𝑎 𝑛 𝑛 superscript 𝑧 𝑎 𝑛 Kummer-confluent-hypergeometric-M 𝑛 1 𝑎 𝑛 𝑧 superscript 𝑧 𝑎 superscript subscript 𝑠 0 𝑛 binomial 𝑛 𝑠 Pochhammer 𝑎 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-% n\right)_{n}}}{z^{a+n}}M\left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(% }{)}{0.0pt}{}{n}{s}{\left(a\right)_{s}}z^{-s}.}}
    0 references
    DLMF id
    DLMF:13.2.E8
    0 references
    Symbols used
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ 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.m2addec
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    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.m2abdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1addec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1aadec
    0 references
     
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