DLMF:17.9.E12 (Q5438)

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DLMF:17.9.E12
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    Statements

    Defining formula
    ϕ 2 3 ⁡ ( a , b , c d , e ; q , d ⁢ e a ⁢ b ⁢ c ) = ( e / b , e / c , c ⁢ q / a , q / d ; q ) ∞ ( e , c ⁢ q / d , q / a , e / ( b ⁢ c ) ; q ) ∞ ⁢ ϕ 2 3 ⁡ ( c , d / a , c ⁢ q / e c ⁢ q / a , b ⁢ c ⁢ q / e ; q , b ⁢ q d ) - ( q / d , e ⁢ q / d , b , c , d / a , d ⁢ e / ( b ⁢ c ⁢ q ) , b ⁢ c ⁢ q 2 / ( d ⁢ e ) ; q ) ∞ ( d / q , e , b ⁢ q / d , c ⁢ q / d , q / a , e / ( b ⁢ c ) , b ⁢ c ⁢ q / e ; q ) ∞ ⁢ ϕ 2 3 ⁡ ( a ⁢ q / d , b ⁢ q / d , c ⁢ q / d q 2 / d , e ⁢ q / d ; q , d ⁢ e a ⁢ b ⁢ c ) , q-hypergeometric-rphis 3 2 𝑎 𝑏 𝑐 𝑑 𝑒 𝑞 𝑑 𝑒 𝑎 𝑏 𝑐 q-multiple-Pochhammer 𝑒 𝑏 𝑒 𝑐 𝑐 𝑞 𝑎 𝑞 𝑑 𝑞 q-multiple-Pochhammer 𝑒 𝑐 𝑞 𝑑 𝑞 𝑎 𝑒 𝑏 𝑐 𝑞 q-hypergeometric-rphis 3 2 𝑐 𝑑 𝑎 𝑐 𝑞 𝑒 𝑐 𝑞 𝑎 𝑏 𝑐 𝑞 𝑒 𝑞 𝑏 𝑞 𝑑 q-multiple-Pochhammer 𝑞 𝑑 𝑒 𝑞 𝑑 𝑏 𝑐 𝑑 𝑎 𝑑 𝑒 𝑏 𝑐 𝑞 𝑏 𝑐 superscript 𝑞 2 𝑑 𝑒 𝑞 q-multiple-Pochhammer 𝑑 𝑞 𝑒 𝑏 𝑞 𝑑 𝑐 𝑞 𝑑 𝑞 𝑎 𝑒 𝑏 𝑐 𝑏 𝑐 𝑞 𝑒 𝑞 q-hypergeometric-rphis 3 2 𝑎 𝑞 𝑑 𝑏 𝑞 𝑑 𝑐 𝑞 𝑑 superscript 𝑞 2 𝑑 𝑒 𝑞 𝑑 𝑞 𝑑 𝑒 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{{}_{3}\phi_{2}}\left({a,b,c\atop d,e};q,\frac{de}% {abc}\right)=\frac{\left(e/b,e/c,cq/a,q/d;q\right)_{\infty}}{\left(e,cq/d,q/a,% e/(bc);q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c,d/a,cq/e\atop cq/a,bcq/e};q,% \frac{bq}{d}\right)-\frac{\left(q/d,eq/d,b,c,d/a,de/(bcq),bcq^{2}/(de);q\right% )_{\infty}}{\left(d/q,e,bq/d,cq/d,q/a,e/(bc),bcq/e;q\right)_{\infty}}{{}_{3}% \phi_{2}}\left({aq/d,bq/d,cq/d\atop q^{2}/d,eq/d};q,\frac{de}{abc}\right),}}
    0 references
    DLMF id
    DLMF:17.9.E12
    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.m2akdec
    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.m6addec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1kdec
    0 references
     
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