DLMF:17.11.E4 (Q5456): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11711 / rank
 
Normal rank
Property / Symbols used: Q11711 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11711 / qualifier
 
xml-id: C17.S1.XMD4.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11719 / rank
 
Normal rank
Property / Symbols used: Q11719 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11719 / qualifier
 
xml-id: C17.S1.XMD8.m1cdec

Latest revision as of 15:10, 2 January 2020

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DLMF:17.11.E4
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    Statements

    Defining formula
    ∑ m 1 , … , m n ≧ 0 ( a ; q ) m 1 + m 2 + ⋯ + m n ⁢ ( b 1 ; q ) m 1 ⁢ ( b 2 ; q ) m 2 ⁢ ⋯ ⁢ ( b n ; q ) m n ⁢ x 1 m 1 ⁢ x 2 m 2 ⁢ ⋯ ⁢ x n m n ( q ; q ) m 1 ⁢ ( q ; q ) m 2 ⁢ ⋯ ⁢ ( q ; q ) m n ⁢ ( c ; q ) m 1 + m 2 + ⋯ + m n = ( a , b 1 ⁢ x 1 , b 2 ⁢ x 2 , … , b n ⁢ x n ; q ) ∞ ( c , x 1 , x 2 , … , x n ; q ) ∞ ⁢ ϕ n n + 1 ⁡ ( c / a , x 1 , x 2 , … , x n b 1 ⁢ x 1 , b 2 ⁢ x 2 , … , b n ⁢ x n ; q , a ) . subscript subscript 𝑚 1 … subscript 𝑚 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 subscript 𝑚 1 subscript 𝑚 2 ⋯ subscript 𝑚 𝑛 q-Pochhammer-symbol subscript 𝑏 1 𝑞 subscript 𝑚 1 q-Pochhammer-symbol subscript 𝑏 2 𝑞 subscript 𝑚 2 ⋯ q-Pochhammer-symbol subscript 𝑏 𝑛 𝑞 subscript 𝑚 𝑛 superscript subscript 𝑥 1 subscript 𝑚 1 superscript subscript 𝑥 2 subscript 𝑚 2 ⋯ superscript subscript 𝑥 𝑛 subscript 𝑚 𝑛 q-Pochhammer-symbol 𝑞 𝑞 subscript 𝑚 1 q-Pochhammer-symbol 𝑞 𝑞 subscript 𝑚 2 ⋯ q-Pochhammer-symbol 𝑞 𝑞 subscript 𝑚 𝑛 q-Pochhammer-symbol 𝑐 𝑞 subscript 𝑚 1 subscript 𝑚 2 ⋯ subscript 𝑚 𝑛 q-multiple-Pochhammer 𝑎 subscript 𝑏 1 subscript 𝑥 1 subscript 𝑏 2 subscript 𝑥 2 … subscript 𝑏 𝑛 subscript 𝑥 𝑛 𝑞 q-multiple-Pochhammer 𝑐 subscript 𝑥 1 subscript 𝑥 2 … subscript 𝑥 𝑛 𝑞 q-hypergeometric-rphis 𝑛 1 𝑛 𝑐 𝑎 subscript 𝑥 1 subscript 𝑥 2 … subscript 𝑥 𝑛 subscript 𝑏 1 subscript 𝑥 1 subscript 𝑏 2 subscript 𝑥 2 … subscript 𝑏 𝑛 subscript 𝑥 𝑛 𝑞 𝑎 {\displaystyle{\displaystyle\sum_{m_{1},\dots,m_{n}\geqq 0}\frac{\left(a;q% \right)_{m_{1}+m_{2}+\cdots+m_{n}}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q% \right)_{m_{2}}\cdots\left(b_{n};q\right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}% \cdots x_{n}^{m_{n}}}{\left(q;q\right)_{m_{1}}\left(q;q\right)_{m_{2}}\cdots% \left(q;q\right)_{m_{n}}\left(c;q\right)_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{% \left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x_{n};q\right)_{\infty}}{\left(c,x_{1% },x_{2},\dots,x_{n};q\right)_{\infty}}{{}_{n+1}\phi_{n}}\left({c/a,x_{1},x_{2}% ,\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x_{n}};q,a\right).}}
    0 references
    DLMF id
    DLMF:17.11.E4
    0 references
    Symbols used
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2abdec
    0 references
    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.m2aadec
    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.m6acdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1cdec
    0 references
    Q11716
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C17.S1.XMD3.m1dec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1bdec
    0 references
    Q11719
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C17.S1.XMD8.m1cdec
    0 references
     
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