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DLMF:16.11.E11 - MaRDI portal

DLMF:16.11.E11 (Q5942)

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DLMF:16.11.E11
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    Statements

    DLMF defining formula
    F q p ⁡ ( a 1 + r , … , a p + r b 1 + r , … , b q + r ; z ) = ∑ n = 0 m - 1 ( a 1 + r ) n ⁢ ⋯ ⁢ ( a p + r ) n ( b 1 + r ) n ⁢ ⋯ ⁢ ( b q + r ) n ⁢ z n n ! + O ⁡ ( 1 r ( q - p ) ⁢ m ) , Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 𝑟 … subscript 𝑎 𝑝 𝑟 subscript 𝑏 1 𝑟 … subscript 𝑏 𝑞 𝑟 𝑧 superscript subscript 𝑛 0 𝑚 1 Pochhammer subscript 𝑎 1 𝑟 𝑛 ⋯ Pochhammer subscript 𝑎 𝑝 𝑟 𝑛 Pochhammer subscript 𝑏 1 𝑟 𝑛 ⋯ Pochhammer subscript 𝑏 𝑞 𝑟 𝑛 superscript 𝑧 𝑛 𝑛 Big-O 1 superscript 𝑟 𝑞 𝑝 𝑚 {\displaystyle{\displaystyle{{}_{p}F_{q}}\left({a_{1}+r,\dots,a_{p}+r\atop b_{% 1}+r,\dots,b_{q}+r};z\right)=\sum_{n=0}^{m-1}\frac{{\left(a_{1}+r\right)_{n}}% \cdots{\left(a_{p}+r\right)_{n}}}{{\left(b_{1}+r\right)_{n}}\cdots{\left(b_{q}% +r\right)_{n}}}\frac{z^{n}}{n!}+O\left(\frac{1}{r^{(q-p)m}}\right),}}
    0 references
    Symbols used
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
    0 references
    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.m1aedec
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    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
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    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    nonnegative integer
    DLMF defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1fdec
    0 references
    nonnegative integer
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1hdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1gdec
    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.m1hdec
    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.m1hdec
    0 references
    large parameter
    DLMF defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C16.S11.XMD6.m1adec
    0 references

    Identifiers

     
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