DLMF:17.4.E2 (Q5351)

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DLMF:17.4.E2
No description defined

    Statements

    Defining formula
    lim q → 1 - ⁡ ϕ r r + 1 ⁡ ( q a 0 , q a 1 , … , q a r q b 1 , … , q b r ; q , z ) = F r r + 1 ⁡ ( a 0 , a 1 , … , a r b 1 , … , b r ; z ) . subscript → 𝑞 limit-from 1 q-hypergeometric-rphis 𝑟 1 𝑟 superscript 𝑞 subscript 𝑎 0 superscript 𝑞 subscript 𝑎 1 … superscript 𝑞 subscript 𝑎 𝑟 superscript 𝑞 subscript 𝑏 1 … superscript 𝑞 subscript 𝑏 𝑟 𝑞 𝑧 Gauss-hypergeometric-pFq 𝑟 1 𝑟 subscript 𝑎 0 subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑟 𝑧 {\displaystyle{\displaystyle\lim_{q\to 1-}{{}_{r+1}\phi_{r}}\left({q^{a_{0}},q% ^{a_{1}},\dots,q^{a_{r}}\atop q^{b_{1}},\dots,q^{b_{r}}};q,z\right)={{}_{r+1}F% _{r}}\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},\dots,b_{r}};z\right).}}
    0 references
    DLMF id
    DLMF:17.4.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.m1adec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    Defining formula
    ϕ s r + 1 ⁡ ( a 0 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rphis 𝑟 1 𝑠 subscript 𝑎 0 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E1.m2aadec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1adec
    0 references
    Q11714
    Defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C17.S1.XMD5.m1adec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1adec
    0 references
     
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