DLMF:17.10.E6 (Q5452): Difference between revisions

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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.m2aadec
Property / Symbols used
 
Property / Symbols used: bilateral basic hypergeometric (or bilateral $$q$$ -hypergeometric) function / rank
 
Normal rank
Property / Symbols used: bilateral basic hypergeometric (or bilateral $$q$$ -hypergeometric) function / qualifier
 
Defining formula:

ψ s r ⁡ ( a 1 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rpsis 𝑟 𝑠 subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}

\qgenhyperpsi{\NVar{r}}{\NVar{s}}@{\NVar{a_{1},\dots,a_{r}}}{\NVar{b_{1},\dots,b_{s}}}{\NVar{q}}{\NVar{z}}
Property / Symbols used: bilateral basic hypergeometric (or bilateral $$q$$ -hypergeometric) function / qualifier
 
xml-id: C17.S4.E3.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11713 / rank
 
Normal rank
Property / Symbols used: Q11713 / qualifier
 
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}}}}

\qmultiPochhammersym{\NVar{a_{1},a_{2},\dots,a_{r}}}{\NVar{q}}{\NVar{n}}
Property / Symbols used: Q11713 / qualifier
 
xml-id: C17.S2.SS1.p1.m6aedec
Property / Symbols used
 
Property / Symbols used: Q11715 / rank
 
Normal rank
Property / Symbols used: Q11715 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11715 / qualifier
 
xml-id: C17.S1.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q11710 / rank
 
Normal rank
Property / Symbols used: Q11710 / qualifier
 
Defining formula:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11710 / qualifier
 
xml-id: C17.S1.XMD10.m1edec

Latest revision as of 15:09, 2 January 2020

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DLMF:17.10.E6
No description defined

    Statements

    Defining formula
    ( a ⁢ q / b , a ⁢ q / c , a ⁢ q / d , a ⁢ q / e , a ⁢ q / f , q / ( a ⁢ b ) , q / ( a ⁢ c ) , q / ( a ⁢ d ) , q / ( a ⁢ e ) , q / ( a ⁢ f ) ; q ) ∞ ( a ⁢ g , a ⁢ h , a ⁢ k , g / a , h / a , k / a , q ⁢ a 2 , q / a 2 ; q ) ∞ ⁢ ψ 10 10 ⁡ ( q ⁢ a , - q ⁢ a , b ⁢ a , c ⁢ a , d ⁢ a , e ⁢ a , f ⁢ a , g ⁢ a , h ⁢ a , k ⁢ a a , - a , a ⁢ q / b , a ⁢ q / c , a ⁢ q / d , a ⁢ q / e , a ⁢ q / f , a ⁢ q / g , a ⁢ q / h , a ⁢ q / k ; q , q 2 b ⁢ c ⁢ d ⁢ e ⁢ f ⁢ g ⁢ h ⁢ k ) = ( q , q / ( b ⁢ g ) , q / ( c ⁢ g ) , q / ( d ⁢ g ) , q / ( e ⁢ g ) , q / ( f ⁢ g ) , q ⁢ g / b , q ⁢ g / c , q ⁢ g / d , q ⁢ g / e , q ⁢ g / f ; q ) ∞ ( g ⁢ h , g ⁢ k , h / g , a ⁢ g , q / ( a ⁢ g ) , g / a , a ⁢ q / g , q ⁢ g 2 ; q ) ∞ ⁢ ϕ 9 10 ⁡ ( g 2 , q ⁢ g , - q ⁢ g , g ⁢ b , g ⁢ c , g ⁢ d , g ⁢ e , g ⁢ f , g ⁢ h , g ⁢ k g , - g , q ⁢ g / b , q ⁢ g / c , q ⁢ g / d , q ⁢ g / e , q ⁢ g / f , q ⁢ g / h , q ⁢ g / k ; q , q 2 b ⁢ c ⁢ d ⁢ e ⁢ f ⁢ g ⁢ h ⁢ k ) + idem ⁡ ( g ; h , k ) . q-multiple-Pochhammer 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑎 𝑞 𝑑 𝑎 𝑞 𝑒 𝑎 𝑞 𝑓 𝑞 𝑎 𝑏 𝑞 𝑎 𝑐 𝑞 𝑎 𝑑 𝑞 𝑎 𝑒 𝑞 𝑎 𝑓 𝑞 q-multiple-Pochhammer 𝑎 𝑔 𝑎 ℎ 𝑎 𝑘 𝑔 𝑎 ℎ 𝑎 𝑘 𝑎 𝑞 superscript 𝑎 2 𝑞 superscript 𝑎 2 𝑞 q-hypergeometric-rpsis 10 10 𝑞 𝑎 𝑞 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 𝑎 𝑒 𝑎 𝑓 𝑎 𝑔 𝑎 ℎ 𝑎 𝑘 𝑎 𝑎 𝑎 𝑎 𝑞 𝑏 𝑎 𝑞 𝑐 𝑎 𝑞 𝑑 𝑎 𝑞 𝑒 𝑎 𝑞 𝑓 𝑎 𝑞 𝑔 𝑎 𝑞 ℎ 𝑎 𝑞 𝑘 𝑞 superscript 𝑞 2 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑘 q-multiple-Pochhammer 𝑞 𝑞 𝑏 𝑔 𝑞 𝑐 𝑔 𝑞 𝑑 𝑔 𝑞 𝑒 𝑔 𝑞 𝑓 𝑔 𝑞 𝑔 𝑏 𝑞 𝑔 𝑐 𝑞 𝑔 𝑑 𝑞 𝑔 𝑒 𝑞 𝑔 𝑓 𝑞 q-multiple-Pochhammer 𝑔 ℎ 𝑔 𝑘 ℎ 𝑔 𝑎 𝑔 𝑞 𝑎 𝑔 𝑔 𝑎 𝑎 𝑞 𝑔 𝑞 superscript 𝑔 2 𝑞 q-hypergeometric-rphis 10 9 superscript 𝑔 2 𝑞 𝑔 𝑞 𝑔 𝑔 𝑏 𝑔 𝑐 𝑔 𝑑 𝑔 𝑒 𝑔 𝑓 𝑔 ℎ 𝑔 𝑘 𝑔 𝑔 𝑞 𝑔 𝑏 𝑞 𝑔 𝑐 𝑞 𝑔 𝑑 𝑞 𝑔 𝑒 𝑞 𝑔 𝑓 𝑞 𝑔 ℎ 𝑞 𝑔 𝑘 𝑞 superscript 𝑞 2 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑘 idem 𝑔 ℎ 𝑘 {\displaystyle{\displaystyle\frac{\left(aq/b,aq/c,aq/d,aq/e,aq/f,q/(ab),q/(ac)% ,q/(ad),q/(ae),q/(af);q\right)_{\infty}}{\left(ag,ah,ak,g/a,h/a,k/a,qa^{2},q/a% ^{2};q\right)_{\infty}}\*{{}_{10}\psi_{10}}\left({qa,-qa,ba,ca,da,ea,fa,ga,ha,% ka\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,aq/k};q,\frac{q^{2}}{bcdefghk}% \right)=\frac{\left(q,q/(bg),q/(cg),q/(dg),q/(eg),q/(fg),qg/b,qg/c,qg/d,qg/e,% qg/f;q\right)_{\infty}}{\left(gh,gk,h/g,ag,q/(ag),g/a,aq/g,qg^{2};q\right)_{% \infty}}\*{{}_{10}\phi_{9}}\left({g^{2},qg,-qg,gb,gc,gd,ge,gf,gh,gk\atop g,-g,% qg/b,qg/c,qg/d,qg/e,qg/f,qg/h,qg/k};q,\frac{q^{2}}{bcdefghk}\right)+\mathrm{% idem}\left(g;h,k\right).}}
    0 references
    DLMF id
    DLMF:17.10.E6
    0 references
    Symbols used
    Q11721
    Defining formula
    idem ⁡ ( χ 1 ; χ 2 , … , χ n ) idem subscript 𝜒 1 subscript 𝜒 2 … subscript 𝜒 𝑛 {\displaystyle{\displaystyle\mathrm{idem}\left(\NVar{\chi_{1}};\NVar{\chi_{2},% \dots,\chi_{n}}\right)}}
    xml-id
    C17.S1.p3.m2aadec
    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
    bilateral basic hypergeometric (or bilateral $$q$$ -hypergeometric) function
    Defining formula
    ψ s r ⁡ ( a 1 , … , a r ; b 1 , … , b s ; q , z ) q-hypergeometric-rpsis 𝑟 𝑠 subscript 𝑎 1 … subscript 𝑎 𝑟 subscript 𝑏 1 … subscript 𝑏 𝑠 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},% \dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
    xml-id
    C17.S4.E3.m2aedec
    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.m6aedec
    0 references
    Q11715
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C17.S1.XMD1.m1dec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1edec
    0 references
     
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