DLMF:17.9.E16 (Q5442)

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DLMF:17.9.E16
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    Statements

    test
    ϕ 7 8 ⁡ ( a , q ⁢ a 1 2 , - q ⁢ a 1 2 , b , c , d , e , f a 1 2 , - a 1 2 , a ⁢ q / b , a ⁢ q / c , a ⁢ q / d , a ⁢ q / e , a ⁢ q / f ; q , a 2 ⁢ q 2 b ⁢ c ⁢ d ⁢ e ⁢ f ) = ( a ⁢ q , a ⁢ q / ( d ⁢ e ) , a ⁢ q / ( d ⁢ f ) , a ⁢ q / ( e ⁢ f ) ; q ) ∞ ( a ⁢ q / d , a ⁢ q / e , a ⁢ q / f , a ⁢ q / ( d ⁢ e ⁢ f ) ; q ) ∞ ⁢ ϕ 3 4 ⁡ ( a ⁢ q / ( b ⁢ c ) , d , e , f a ⁢ q / b , a ⁢ q / c , d ⁢ e ⁢ f / a ; q , q ) + ( a ⁢ q , a ⁢ q / ( b ⁢ c ) , d , e , f , a 2 ⁢ q 2 / ( b ⁢ d ⁢ e ⁢ f ) , a 2 ⁢ q 2 / ( c ⁢ d ⁢ e ⁢ f ) ; q ) ∞ ( a ⁢ q / b , a ⁢ q / c , a ⁢ q / d , a ⁢ q / e , a ⁢ q / f , a 2 ⁢ q 2 / ( b ⁢ c ⁢ d ⁢ e ⁢ f ) , d ⁢ e ⁢ f / ( a ⁢ q ) ; q ) ∞ ⁢ ϕ 3 4 ⁡ ( a ⁢ q / ( d ⁢ e ) , a ⁢ q / ( d ⁢ f ) , a ⁢ q / ( e ⁢ f ) , a 2 ⁢ q 2 / ( b ⁢ c ⁢ d ⁢ e ⁢ f ) a 2 ⁢ q 2 / ( b ⁢ d ⁢ e ⁢ f ) , a 2 ⁢ q 2 / ( c ⁢ d ⁢ e ⁢ f ) , a ⁢ q 2 / ( d ⁢ e ⁢ f ) ; q , q ) . q-hypergeometric-rphis 8 7 𝑎 𝑞 superscript 𝑎 1 2 𝑞 superscript 𝑎 1 2 𝑏 𝑐 𝑑 𝑒 𝑓 superscript 𝑎 1 2 superscript 𝑎 1 2 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑎 𝑞 𝑑 𝑎 𝑞 𝑒 𝑎 𝑞 𝑓 𝑞 superscript 𝑎 2 superscript 𝑞 2 𝑏 𝑐 𝑑 𝑒 𝑓 q-multiple-Pochhammer 𝑎 𝑞 𝑎 𝑞 𝑑 𝑒 𝑎 𝑞 𝑑 𝑓 𝑎 𝑞 𝑒 𝑓 𝑞 q-multiple-Pochhammer 𝑎 𝑞 𝑑 𝑎 𝑞 𝑒 𝑎 𝑞 𝑓 𝑎 𝑞 𝑑 𝑒 𝑓 𝑞 q-hypergeometric-rphis 4 3 𝑎 𝑞 𝑏 𝑐 𝑑 𝑒 𝑓 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑑 𝑒 𝑓 𝑎 𝑞 𝑞 q-multiple-Pochhammer 𝑎 𝑞 𝑎 𝑞 𝑏 𝑐 𝑑 𝑒 𝑓 superscript 𝑎 2 superscript 𝑞 2 𝑏 𝑑 𝑒 𝑓 superscript 𝑎 2 superscript 𝑞 2 𝑐 𝑑 𝑒 𝑓 𝑞 q-multiple-Pochhammer 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑎 𝑞 𝑑 𝑎 𝑞 𝑒 𝑎 𝑞 𝑓 superscript 𝑎 2 superscript 𝑞 2 𝑏 𝑐 𝑑 𝑒 𝑓 𝑑 𝑒 𝑓 𝑎 𝑞 𝑞 q-hypergeometric-rphis 4 3 𝑎 𝑞 𝑑 𝑒 𝑎 𝑞 𝑑 𝑓 𝑎 𝑞 𝑒 𝑓 superscript 𝑎 2 superscript 𝑞 2 𝑏 𝑐 𝑑 𝑒 𝑓 superscript 𝑎 2 superscript 𝑞 2 𝑏 𝑑 𝑒 𝑓 superscript 𝑎 2 superscript 𝑞 2 𝑐 𝑑 𝑒 𝑓 𝑎 superscript 𝑞 2 𝑑 𝑒 𝑓 𝑞 𝑞 {\displaystyle{\displaystyle{{}_{8}\phi_{7}}\left({a,qa^{\frac{1}{2}},-qa^{% \frac{1}{2}},b,c,d,e,f\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq% /e,aq/f};q,\frac{a^{2}q^{2}}{bcdef}\right)=\frac{\left(aq,aq/(de),aq/(df),aq/(% ef);q\right)_{\infty}}{\left(aq/d,aq/e,aq/f,aq/(def);q\right)_{\infty}}{{}_{4}% \phi_{3}}\left({aq/(bc),d,e,f\atop aq/b,aq/c,def/a};q,q\right)+\frac{\left(aq,% aq/(bc),d,e,f,a^{2}q^{2}/(bdef),a^{2}q^{2}/(cdef);q\right)_{\infty}}{\left(aq/% b,aq/c,aq/d,aq/e,aq/f,a^{2}q^{2}/(bcdef),def/(aq);q\right)_{\infty}}\*{{}_{4}% \phi_{3}}\left({aq/(de),aq/(df),aq/(ef),a^{2}q^{2}/(bcdef)\atop a^{2}q^{2}/(% bdef),a^{2}q^{2}/(cdef),aq^{2}/(def)};q,q\right).}}
    0 references
    DLMF id
    DLMF:17.9.E16
    0 references
    Symbols used
    basic hypergeometric (or $$q$$ -hypergeometric) function
    test
    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
    Q11713
    test
    ( 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.m6ahdec
    0 references
     
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