DLMF:18.33.E13 (Q6021)

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DLMF:18.33.E13
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    Statements

    Defining formula
    ϕ n ⁢ ( z ) = ∑ ℓ = 0 n ( λ + 1 ) ℓ ⁢ ( λ ) n - ℓ ℓ ! ⁢ ( n - ℓ ) ! ⁢ z ℓ = ( λ ) n n ! ⁢ F 1 2 ⁡ ( - n , λ + 1 - λ - n + 1 ; z ) , subscript italic-ϕ 𝑛 𝑧 superscript subscript ℓ 0 𝑛 Pochhammer 𝜆 1 ℓ Pochhammer 𝜆 𝑛 ℓ ℓ 𝑛 ℓ superscript 𝑧 ℓ Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝜆 1 𝜆 𝑛 1 𝑧 {\displaystyle{\displaystyle\phi_{n}(z)=\sum_{\ell=0}^{n}\frac{{\left(\lambda+% 1\right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\,(n-\ell)!}\,z^{\ell}=% \frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda+1\atop-% \lambda-n+1};z\right),}}
    0 references
    DLMF id
    DLMF:18.33.E13
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1kdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1bdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1kdec
    0 references
    Q11791
    Defining formula
    ϕ n ⁢ ( z ) subscript italic-ϕ 𝑛 𝑧 {\displaystyle{\displaystyle\phi_{n}(z)}}
    xml-id
    C18.S33.XMD1.m1kdec
    0 references
     
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