DLMF:16.10.E1 (Q5231)

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DLMF:16.10.E1
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    Statements

    Defining formula
    F q + s p + r ⁡ ( a 1 , … , a p , c 1 , … , c r b 1 , … , b q , d 1 , … , d s ; z ⁢ ζ ) = ∑ k = 0 ∞ ( 𝐚 ) k ⁢ ( α ) k ⁢ ( β ) k ⁢ ( - z ) k ( 𝐛 ) k ⁢ ( γ + k ) k ⁢ k ! ⁢ F q + 1 p + 2 ⁡ ( α + k , β + k , a 1 + k , … , a p + k γ + 2 ⁢ k + 1 , b 1 + k , … , b q + k ; z ) ⁢ F s + 2 r + 2 ⁡ ( - k , γ + k , c 1 , … , c r α , β , d 1 , … , d s ; ζ ) . Gauss-hypergeometric-pFq 𝑝 𝑟 𝑞 𝑠 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑐 1 … subscript 𝑐 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑞 subscript 𝑑 1 … subscript 𝑑 𝑠 𝑧 𝜁 superscript subscript 𝑘 0 Pochhammer 𝐚 𝑘 Pochhammer 𝛼 𝑘 Pochhammer 𝛽 𝑘 superscript 𝑧 𝑘 Pochhammer 𝐛 𝑘 Pochhammer 𝛾 𝑘 𝑘 𝑘 Gauss-hypergeometric-pFq 𝑝 2 𝑞 1 𝛼 𝑘 𝛽 𝑘 subscript 𝑎 1 𝑘 … subscript 𝑎 𝑝 𝑘 𝛾 2 𝑘 1 subscript 𝑏 1 𝑘 … subscript 𝑏 𝑞 𝑘 𝑧 Gauss-hypergeometric-pFq 𝑟 2 𝑠 2 𝑘 𝛾 𝑘 subscript 𝑐 1 … subscript 𝑐 𝑟 𝛼 𝛽 subscript 𝑑 1 … subscript 𝑑 𝑠 𝜁 {\displaystyle{\displaystyle{{}_{p+r}F_{q+s}}\left({a_{1},\dots,a_{p},c_{1},% \dots,c_{r}\atop b_{1},\dots,b_{q},d_{1},\dots,d_{s}};z\zeta\right)=\sum_{k=0}% ^{\infty}\frac{{\left(\mathbf{a}\right)_{k}}{\left(\alpha\right)_{k}}{\left(% \beta\right)_{k}}(-z)^{k}}{{\left(\mathbf{b}\right)_{k}}{\left(\gamma+k\right)% _{k}}k!}{{}_{p+2}F_{q+1}}\left({\alpha+k,\beta+k,a_{1}+k,\dots,a_{p}+k\atop% \gamma+2k+1,b_{1}+k,\dots,b_{q}+k};z\right){{}_{r+2}F_{s+2}}\left({-k,\gamma+k% ,c_{1},\dots,c_{r}\atop\alpha,\beta,d_{1},\dots,d_{s}};\zeta\right).}}
    0 references
    DLMF id
    DLMF:16.10.E1
    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.m1adec
    0 references
     
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