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DLMF:17.7.E18 - MaRDI portal

DLMF:17.7.E18 (Q6111)

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DLMF:17.7.E18
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    Statements

    DLMF defining formula
    ϕ r + 1 r + 2 ⁡ ( a , b , b 1 ⁢ q m 1 , … , b r ⁢ q m r b ⁢ q , b 1 , … , b r ; q , a - 1 ⁢ q 1 - ( m 1 + ⋯ + m r ) ) = ( q , b ⁢ q / a ; q ) ∞ ⁢ ( b 1 / b ; q ) m 1 ⁢ ⋯ ⁢ ( b r / b ; q ) m r ( b ⁢ q , q / a ; q ) ∞ ⁢ ( b 1 ; q ) m 1 ⁢ ⋯ ⁢ ( b r ; q ) m r ⁢ b m 1 + ⋯ + m r , q-hypergeometric-rphis 𝑟 2 𝑟 1 𝑎 𝑏 subscript 𝑏 1 superscript 𝑞 subscript 𝑚 1 … subscript 𝑏 𝑟 superscript 𝑞 subscript 𝑚 𝑟 𝑏 𝑞 subscript 𝑏 1 … subscript 𝑏 𝑟 𝑞 superscript 𝑎 1 superscript 𝑞 1 subscript 𝑚 1 ⋯ subscript 𝑚 𝑟 q-multiple-Pochhammer 𝑞 𝑏 𝑞 𝑎 𝑞 q-Pochhammer-symbol subscript 𝑏 1 𝑏 𝑞 subscript 𝑚 1 ⋯ q-Pochhammer-symbol subscript 𝑏 𝑟 𝑏 𝑞 subscript 𝑚 𝑟 q-multiple-Pochhammer 𝑏 𝑞 𝑞 𝑎 𝑞 q-Pochhammer-symbol subscript 𝑏 1 𝑞 subscript 𝑚 1 ⋯ q-Pochhammer-symbol subscript 𝑏 𝑟 𝑞 subscript 𝑚 𝑟 superscript 𝑏 subscript 𝑚 1 ⋯ subscript 𝑚 𝑟 {\displaystyle{\displaystyle{{}_{r+2}\phi_{r+1}}\left({a,b,b_{1}q^{m_{1}},% \dots,b_{r}q^{m_{r}}\atop bq,b_{1},\dots,b_{r}};q,a^{-1}q^{1-(m_{1}+\cdots+m_{% r})}\right)=\frac{\left(q,bq/a;q\right)_{\infty}\left(b_{1}/b;q\right)_{m_{1}}% \cdots\left(b_{r}/b;q\right)_{m_{r}}}{\left(bq,q/a;q\right)_{\infty}\left(b_{1% };q\right)_{m_{1}}\cdots\left(b_{r};q\right)_{m_{r}}}b^{m_{1}+\cdots+m_{r}},}}
    0 references
    Symbols used
    Q11269
    DLMF defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2abdec
    0 references
    basic hypergeometric (or $$q$$ -hypergeometric) function
    DLMF 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.m2aqdec
    0 references
    Q11684
    DLMF defining formula
    ( a 1 , a 2 , … , a r ; q ) n q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}% \right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m6aqdec
    0 references
    complex base
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1qdec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C17.S1.XMD3.m1dec
    0 references
    nonnegative integer
    DLMF defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C17.S1.XMD5.m1dec
    0 references

    Identifiers

     
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