DLMF:18.26.E21 (Q5951)

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DLMF:18.26.E21
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    Statements

    Defining formula
    ( 1 - z ) y ⁢ F 1 2 ⁡ ( y - N , y + γ + 1 - δ - N ; z ) = ∑ n = 0 N ( γ + 1 ) n ⁢ ( - N ) n ( - δ - N ) n ⁢ n ! ⁢ R n ⁡ ( y ⁢ ( y + γ + δ + 1 ) ; γ , δ , N ) ⁢ z n . superscript 1 𝑧 𝑦 Gauss-hypergeometric-F-as-2F1 𝑦 𝑁 𝑦 𝛾 1 𝛿 𝑁 𝑧 superscript subscript 𝑛 0 𝑁 Pochhammer 𝛾 1 𝑛 Pochhammer 𝑁 𝑛 Pochhammer 𝛿 𝑁 𝑛 𝑛 dual-Hahn-R 𝑛 𝑦 𝑦 𝛾 𝛿 1 𝛾 𝛿 𝑁 superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-% \delta-N};z\right)=\sum_{n=0}^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N% \right)_{n}}}{{\left(-\delta-N\right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1)% ;\gamma,\delta,N\right)z^{n}.}}
    0 references
    DLMF id
    DLMF:18.26.E21
    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.m5acdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1afdec
    0 references
    Q11769
    Defining formula
    R n ⁡ ( x ; γ , δ , N ) dual-Hahn-R 𝑛 𝑥 𝛾 𝛿 𝑁 {\displaystyle{\displaystyle R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{% \delta},\NVar{N}\right)}}
    xml-id
    C18.S25.T1.t1.r5.m2aedec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5afdec
    0 references
     
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