DLMF:17.6.E16 (Q5383)

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DLMF:17.6.E16
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    Statements

    Defining formula
    ϕ 1 2 ⁡ ( a , b c ; q , z ) = ( b , c / a , a ⁢ z , q / ( a ⁢ z ) ; q ) ∞ ( c , b / a , z , q / z ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( a , a ⁢ q / c a ⁢ q / b ; q , c ⁢ q / ( a ⁢ b ⁢ z ) ) + ( a , c / b , b ⁢ z , q / ( b ⁢ z ) ; q ) ∞ ( c , a / b , z , q / z ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( b , b ⁢ q / c b ⁢ q / a ; q , c ⁢ q / ( a ⁢ b ⁢ z ) ) , q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑧 q-multiple-Pochhammer 𝑏 𝑐 𝑎 𝑎 𝑧 𝑞 𝑎 𝑧 𝑞 q-multiple-Pochhammer 𝑐 𝑏 𝑎 𝑧 𝑞 𝑧 𝑞 q-hypergeometric-rphis 2 1 𝑎 𝑎 𝑞 𝑐 𝑎 𝑞 𝑏 𝑞 𝑐 𝑞 𝑎 𝑏 𝑧 q-multiple-Pochhammer 𝑎 𝑐 𝑏 𝑏 𝑧 𝑞 𝑏 𝑧 𝑞 q-multiple-Pochhammer 𝑐 𝑎 𝑏 𝑧 𝑞 𝑧 𝑞 q-hypergeometric-rphis 2 1 𝑏 𝑏 𝑞 𝑐 𝑏 𝑞 𝑎 𝑞 𝑐 𝑞 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=% \frac{\left(b,c/a,az,q/(az);q\right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{% \infty}}{{}_{2}\phi_{1}}\left({a,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left% (a,c/b,bz,q/(bz);q\right)_{\infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2% }\phi_{1}}\left({b,bq/c\atop bq/a};q,cq/(abz)\right),}}
    0 references
    DLMF id
    DLMF:17.6.E16
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    constraint
    | a ⁢ b ⁢ z | < | c ⁢ q | 𝑎 𝑏 𝑧 𝑐 𝑞 {\displaystyle{\displaystyle|abz|<|cq|}}
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    Symbols used
    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.m2aodec
    0 references
     
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