DLMF:17.6.E26 (Q5393): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: $$q$$ -differential operator (locally) / rank
 
Normal rank
Property / Symbols used: $$q$$ -differential operator (locally) / qualifier
 
Defining formula:

𝒟 q subscript 𝒟 𝑞 {\displaystyle{\displaystyle\mathcal{D}_{q}}}

\mathcal{D}_{q}
Property / Symbols used: $$q$$ -differential operator (locally) / qualifier
 
xml-id: C17.S2.XMD1.m1adec

Latest revision as of 15:00, 2 January 2020

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DLMF:17.6.E26
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    Statements

    Defining formula
    𝒟 q n ⁢ ( ( z ; q ) ∞ ( a ⁢ b ⁢ z / c ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( a , b c ; q , z ) ) = ( c / a , c / b ; q ) n ( c ; q ) n ⁢ ( 1 - q ) n ⁢ ( a ⁢ b c ) n ⁢ ( z ⁢ q n ; q ) ∞ ( a ⁢ b ⁢ z / c ; q ) ∞ ⁢ ϕ 1 2 ⁡ ( a , b c ⁢ q n ; q , z ⁢ q n ) . superscript subscript 𝒟 𝑞 𝑛 q-Pochhammer-symbol 𝑧 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑧 𝑐 𝑞 q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 𝑞 𝑧 q-multiple-Pochhammer 𝑐 𝑎 𝑐 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑐 𝑞 𝑛 superscript 1 𝑞 𝑛 superscript 𝑎 𝑏 𝑐 𝑛 q-Pochhammer-symbol 𝑧 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑧 𝑐 𝑞 q-hypergeometric-rphis 2 1 𝑎 𝑏 𝑐 superscript 𝑞 𝑛 𝑞 𝑧 superscript 𝑞 𝑛 {\displaystyle{\displaystyle\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{% \infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z% \right)\right)=\frac{\left(c/a,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}% }\left(\frac{ab}{c}\right)^{n}\frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/% c;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).}}
    0 references
    DLMF id
    DLMF:17.6.E26
    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.m2ajdec
    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.m2aydec
    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.m6andec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1ydec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1hdec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1sdec
    0 references
    $$q$$ -differential operator (locally)
    Defining formula
    𝒟 q subscript 𝒟 𝑞 {\displaystyle{\displaystyle\mathcal{D}_{q}}}
    xml-id
    C17.S2.XMD1.m1adec
    0 references
     
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