DLMF:17.6.E9 (Q5376): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: basic hypergeometric (or $$q$$ -hypergeometric) function / rank
 
Normal rank
Property / Symbols used: basic hypergeometric (or $$q$$ -hypergeometric) function / qualifier
 
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)}}

\qgenhyperphi{\NVar{r+1}}{\NVar{s}}@{\NVar{a_{0},\dots,a_{r}}}{\NVar{b_{1},\dots,b_{s}}}{\NVar{q}}{\NVar{z}}
Property / Symbols used: basic hypergeometric (or $$q$$ -hypergeometric) function / qualifier
 
xml-id: C17.S4.E1.m2ahdec

Revision as of 14:58, 2 January 2020

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DLMF:17.6.E9
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    Statements

    Defining formula
    ϕ 1 2 ⁡ ( q , a ⁢ q b ⁢ q ; q , z ) = - ( 1 - b ) ⁢ ( a ⁢ q / b ) ( 1 - ( a ⁢ q / b ) ) ⁢ ∑ n = 0 ∞ ( a ⁢ q , a ⁢ z ⁢ q / b ; q ) n ⁢ q n ( a ⁢ z ⁢ q 2 / b ; q ) n + ( a ⁢ q , a ⁢ z ⁢ q / b ; q ) ∞ ( a ⁢ q / b ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( q , 0 b ⁢ q ; q , z ) , q-hypergeometric-rphis 2 1 𝑞 𝑎 𝑞 𝑏 𝑞 𝑞 𝑧 1 𝑏 𝑎 𝑞 𝑏 1 𝑎 𝑞 𝑏 superscript subscript 𝑛 0 q-multiple-Pochhammer 𝑎 𝑞 𝑎 𝑧 𝑞 𝑏 𝑞 𝑛 superscript 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 superscript 𝑞 2 𝑏 𝑞 𝑛 q-multiple-Pochhammer 𝑎 𝑞 𝑎 𝑧 𝑞 𝑏 𝑞 q-Pochhammer-symbol 𝑎 𝑞 𝑏 𝑞 q-hypergeometric-rphis 2 1 𝑞 0 𝑏 𝑞 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=-% \frac{(1-b)(aq/b)}{(1-(\ifrac{aq}{b}))}\sum_{n=0}^{\infty}\frac{\left(aq,azq/b% ;q\right)_{n}q^{n}}{\left(azq^{2}/b;q\right)_{n}}+\frac{\left(aq,azq/b;q\right% )_{\infty}}{\left(aq/b;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({q,0\atop bq};q% ,z\right),}}
    0 references
    DLMF id
    DLMF:17.6.E9
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    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.m2aedec
    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.m2ahdec
    0 references
     
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