DLMF:17.6.E15 (Q5382)

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DLMF:17.6.E15
No description defined

    Statements

    Defining formula
    ϕ 1 2 ⁡ ( a , b c ; q , z ) = ( a ⁢ b ⁢ z / c , q / c ; q ) ∞ ( a ⁢ z / c , q / a ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( c / a , c ⁢ q / ( a ⁢ b ⁢ z ) c ⁢ q / ( a ⁢ z ) ; q , b ⁢ q / c ) - ( b , q / c , c / a , a ⁢ z / q , q 2 / ( a ⁢ z ) ; q ) ∞ ( c / q , b ⁢ q / c , q / a , a ⁢ z / c , c ⁢ q / ( a ⁢ z ) ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( a ⁢ q / c , b ⁢ q / c q 2 / c ; q , z ) , q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑧 q-multiple-Pochhammer 𝑎 𝑏 𝑧 𝑐 𝑞 𝑐 𝑞 q-multiple-Pochhammer 𝑎 𝑧 𝑐 𝑞 𝑎 𝑞 q-hypergeometric-rphis 2 1 𝑐 𝑎 𝑐 𝑞 𝑎 𝑏 𝑧 𝑐 𝑞 𝑎 𝑧 𝑞 𝑏 𝑞 𝑐 q-multiple-Pochhammer 𝑏 𝑞 𝑐 𝑐 𝑎 𝑎 𝑧 𝑞 superscript 𝑞 2 𝑎 𝑧 𝑞 q-multiple-Pochhammer 𝑐 𝑞 𝑏 𝑞 𝑐 𝑞 𝑎 𝑎 𝑧 𝑐 𝑐 𝑞 𝑎 𝑧 𝑞 q-hypergeometric-rphis 2 1 𝑎 𝑞 𝑐 𝑏 𝑞 𝑐 superscript 𝑞 2 𝑐 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=% \frac{\left(abz/c,q/c;q\right)_{\infty}}{\left(az/c,q/a;q\right)_{\infty}}{{}_% {2}\phi_{1}}\left({c/a,cq/(abz)\atop cq/(az)};q,bq/c\right)-\frac{\left(b,q/c,% c/a,az/q,q^{2}/(az);q\right)_{\infty}}{\left(c/q,bq/c,q/a,az/c,cq/(az);q\right% )_{\infty}}{{}_{2}\phi_{1}}\left({aq/c,bq/c\atop q^{2}/c};q,z\right),}}
    0 references
    DLMF id
    DLMF:17.6.E15
    0 references
    constraint
    | z | < 1 , | b ⁢ q | < | c | formulae-sequence 𝑧 1 𝑏 𝑞 𝑐 {\displaystyle{\displaystyle|z|<1,|bq|<|c|}}
    0 references
    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.m2andec
    0 references
    Q11713
    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.m6akdec
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    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1ndec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1hdec
    0 references
     
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