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DLMF:17.6.E8 - MaRDI portal

DLMF:17.6.E8 (Q6072)

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DLMF:17.6.E8
No description defined

    Statements

    DLMF defining formula
    ϕ 1 2 ⁡ ( a , b c ; q , z ) = ( a ⁢ b ⁢ z / c ; q ) ∞ ( z ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( c / a , c / b c ; q , a ⁢ b ⁢ z / c ) , q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑧 q-Pochhammer-symbol 𝑎 𝑏 𝑧 𝑐 𝑞 q-Pochhammer-symbol 𝑧 𝑞 q-hypergeometric-rphis 2 1 𝑐 𝑎 𝑐 𝑏 𝑐 𝑞 𝑎 𝑏 𝑧 𝑐 {\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=% \frac{\left(\ifrac{abz}{c};q\right)_{\infty}}{\left(z;q\right)_{\infty}}{{}_{2% }\phi_{1}}\left({c/a,c/b\atop c};q,\ifrac{abz}{c}\right),}}
    0 references
    constraint
    | z | < 1 , | a ⁢ b ⁢ z | < | c | formulae-sequence 𝑧 1 𝑎 𝑏 𝑧 𝑐 {\displaystyle{\displaystyle|z|<1,|abz|<|c|}}
    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.m2addec
    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.m2agdec
    0 references
    complex base
    DLMF defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1gdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1bdec
    0 references

    Identifiers

     
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