DLMF:16.10.E2 (Q5232)

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DLMF:16.10.E2
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    Statements

    Defining formula
    F p p + 1 ⁡ ( a 1 , … , a p + 1 b 1 , … , b p ; z ⁢ ζ ) = ( 1 - z ) - a 1 ⁢ ∑ k = 0 ∞ ( a 1 ) k k ! ⁢ F p p + 1 ⁡ ( - k , a 2 , … , a p + 1 b 1 , … , b p ; ζ ) ⁢ ( z z - 1 ) k . Gauss-hypergeometric-pFq 𝑝 1 𝑝 subscript 𝑎 1 … subscript 𝑎 𝑝 1 subscript 𝑏 1 … subscript 𝑏 𝑝 𝑧 𝜁 superscript 1 𝑧 subscript 𝑎 1 superscript subscript 𝑘 0 Pochhammer subscript 𝑎 1 𝑘 𝑘 Gauss-hypergeometric-pFq 𝑝 1 𝑝 𝑘 subscript 𝑎 2 … subscript 𝑎 𝑝 1 subscript 𝑏 1 … subscript 𝑏 𝑝 𝜁 superscript 𝑧 𝑧 1 𝑘 {\displaystyle{\displaystyle{{}_{p+1}F_{p}}\left({a_{1},\dots,a_{p+1}\atop b_{% 1},\dots,b_{p}};z\zeta\right)=(1-z)^{-a_{1}}\sum_{k=0}^{\infty}\frac{{\left(a_% {1}\right)_{k}}}{k!}{{}_{p+1}F_{p}}\left({-k,a_{2},\dots,a_{p+1}\atop b_{1},% \dots,b_{p}};\zeta\right)\left(\frac{z}{z-1}\right)^{k}.}}
    0 references
    DLMF id
    DLMF:16.10.E2
    0 references
    Symbols used
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1aadec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1adec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1adec
    0 references
     
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