DLMF:17.7.E22 (Q5418)

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DLMF:17.7.E22
No description defined

    Statements

    Defining formula
    ∑ k = - m n ( 1 - a ⁢ d ⁢ p k ⁢ q k ) ⁢ ( 1 - b ⁢ p k / ( d ⁢ q k ) ) ( 1 - a ⁢ d ) ⁢ ( 1 - ( b / d ) ) ⁢ ( a , b ; p ) k ⁢ ( c , a ⁢ d 2 / ( b ⁢ c ) ; q ) k ( d ⁢ q , a ⁢ d ⁢ q / b ; q ) k ⁢ ( a ⁢ d ⁢ p / c , b ⁢ c ⁢ p / d ; p ) k ⁢ q k = ( 1 - a ) ⁢ ( 1 - b ) ⁢ ( 1 - c ) ⁢ ( 1 - ( a ⁢ d 2 / ( b ⁢ c ) ) ) d ⁢ ( 1 - a ⁢ d ) ⁢ ( 1 - ( b / d ) ) ⁢ ( 1 - ( c / d ) ) ⁢ ( 1 - ( a ⁢ d / ( b ⁢ c ) ) ) ⁢ ( ( a ⁢ p , b ⁢ p ; p ) n ⁢ ( c ⁢ q , a ⁢ d 2 ⁢ q / ( b ⁢ c ) ; q ) n ( d ⁢ q , a ⁢ d ⁢ q / b ; q ) n ⁢ ( a ⁢ d ⁢ p / c , b ⁢ c ⁢ p / d ; p ) n - ( c / ( a ⁢ d ) , d / ( b ⁢ c ) ; p ) m + 1 ⁢ ( 1 / d , b / ( a ⁢ d ) ; q ) m + 1 ( 1 / c , b ⁢ c / ( a ⁢ d 2 ) ; q ) m + 1 ⁢ ( 1 / a , 1 / b ; p ) m + 1 ) , superscript subscript 𝑘 𝑚 𝑛 1 𝑎 𝑑 superscript 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑏 superscript 𝑝 𝑘 𝑑 superscript 𝑞 𝑘 1 𝑎 𝑑 1 𝑏 𝑑 q-multiple-Pochhammer 𝑎 𝑏 𝑝 𝑘 q-multiple-Pochhammer 𝑐 𝑎 superscript 𝑑 2 𝑏 𝑐 𝑞 𝑘 q-multiple-Pochhammer 𝑑 𝑞 𝑎 𝑑 𝑞 𝑏 𝑞 𝑘 q-multiple-Pochhammer 𝑎 𝑑 𝑝 𝑐 𝑏 𝑐 𝑝 𝑑 𝑝 𝑘 superscript 𝑞 𝑘 1 𝑎 1 𝑏 1 𝑐 1 𝑎 superscript 𝑑 2 𝑏 𝑐 𝑑 1 𝑎 𝑑 1 𝑏 𝑑 1 𝑐 𝑑 1 𝑎 𝑑 𝑏 𝑐 q-multiple-Pochhammer 𝑎 𝑝 𝑏 𝑝 𝑝 𝑛 q-multiple-Pochhammer 𝑐 𝑞 𝑎 superscript 𝑑 2 𝑞 𝑏 𝑐 𝑞 𝑛 q-multiple-Pochhammer 𝑑 𝑞 𝑎 𝑑 𝑞 𝑏 𝑞 𝑛 q-multiple-Pochhammer 𝑎 𝑑 𝑝 𝑐 𝑏 𝑐 𝑝 𝑑 𝑝 𝑛 q-multiple-Pochhammer 𝑐 𝑎 𝑑 𝑑 𝑏 𝑐 𝑝 𝑚 1 q-multiple-Pochhammer 1 𝑑 𝑏 𝑎 𝑑 𝑞 𝑚 1 q-multiple-Pochhammer 1 𝑐 𝑏 𝑐 𝑎 superscript 𝑑 2 𝑞 𝑚 1 q-multiple-Pochhammer 1 𝑎 1 𝑏 𝑝 𝑚 1 {\displaystyle{\displaystyle\sum_{k=-m}^{n}\frac{(1-adp^{k}q^{k})(1-bp^{k}/(dq% ^{k}))}{(1-ad)(1-(b/d))}\frac{\left(a,b;p\right)_{k}\left(c,ad^{2}/(bc);q% \right)_{k}}{\left(dq,adq/b;q\right)_{k}\left(adp/c,bcp/d;p\right)_{k}}q^{k}=% \frac{(1-a)(1-b)(1-c)(1-(ad^{2}/(bc)))}{d(1-ad)(1-(b/d))(1-(c/d))(1-(ad/(bc)))% }\left(\frac{\left(ap,bp;p\right)_{n}\left(cq,ad^{2}q/(bc);q\right)_{n}}{\left% (dq,adq/b;q\right)_{n}\left(adp/c,bcp/d;p\right)_{n}}-\frac{\left(c/(ad),d/(bc% );p\right)_{m+1}\left(1/d,b/(ad);q\right)_{m+1}}{\left(1/c,bc/(ad^{2});q\right% )_{m+1}\left(1/a,1/b;p\right)_{m+1}}\right),}}
    0 references
    DLMF id
    DLMF:17.7.E22
    0 references
    Symbols used
    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.m6asdec
    0 references
    Q11715
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C17.S1.XMD1.m1bdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1udec
    0 references
    Q11716
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C17.S1.XMD3.m1bdec
    0 references
     
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