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DLMF:16.17.E2 - MaRDI portal

DLMF:16.17.E2 (Q5975)

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DLMF:16.17.E2
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    Statements

    DLMF defining formula
    G p , q m , n ⁡ ( z ; a 1 , … , a p b 1 , … , b q ) = ∑ k = 1 m A p , q , k m , n ⁢ ( z ) ⁢ F q - 1 p ⁡ ( 1 + b k - a 1 , … , 1 + b k - a p 1 + b k - b 1 , … * … , 1 + b k - b q ; ( - 1 ) p - m - n ⁢ z ) , Meijer-G 𝑚 𝑛 𝑝 𝑞 𝑧 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 superscript subscript 𝑘 1 𝑚 superscript subscript 𝐴 𝑝 𝑞 𝑘 𝑚 𝑛 𝑧 Gauss-hypergeometric-pFq 𝑝 𝑞 1 1 subscript 𝑏 𝑘 subscript 𝑎 1 … 1 subscript 𝑏 𝑘 subscript 𝑎 𝑝 1 subscript 𝑏 𝑘 subscript 𝑏 1 … … 1 subscript 𝑏 𝑘 subscript 𝑏 𝑞 superscript 1 𝑝 𝑚 𝑛 𝑧 {\displaystyle{\displaystyle{G^{m,n}_{p,q}}\left(z;{a_{1},\dots,a_{p}\atop b_{% 1},\dots,b_{q}}\right)=\sum_{k=1}^{m}A_{p,q,k}^{m,n}(z){{}_{p}F_{q-1}}\left({1% +b_{k}-a_{1},\dots,1+b_{k}-a_{p}\atop 1+b_{k}-b_{1},\ldots*\dots,1+b_{k}-b_{q}% };(-1)^{p-m-n}z\right),}}
    0 references
    Symbols used
    alternatively F q p ( a ; b ; z ) or F q p ( a b ; z ) generalized hypergeometric function
    DLMF 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
    Meijer $$G$$ -function
    DLMF defining formula
    G p , q m , n ⁡ ( z ; a 1 , … , a p b 1 , … , b q ) Meijer-G 𝑚 𝑛 𝑝 𝑞 𝑧 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle{G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(% \NVar{z};{\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)}}
    xml-id
    C16.S17.E1.m2aadec
    0 references
    nonnegative integer
    DLMF defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1adec
    0 references
    nonnegative integer
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1adec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1adec
    0 references
    real or complex parameters
    DLMF defining formula
    a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
    xml-id
    C16.S1.XMD4.m1adec
    0 references
    real or complex parameters
    DLMF defining formula
    b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
    xml-id
    C16.S1.XMD5.m1adec
    0 references
    coefficient (locally)
    DLMF defining formula
    A p , q m , n ⁢ ( z ) superscript subscript 𝐴 𝑝 𝑞 𝑚 𝑛 𝑧 {\displaystyle{\displaystyle A_{p,q}^{m,n}(z)}}
    xml-id
    C16.S17.XMD1.m1dec
    0 references

    Identifiers

     
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