DLMF:17.6.E11 (Q5378): Difference between revisions

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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.m6agdec
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.m1jdec
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.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11718 / rank
 
Normal rank
Property / Symbols used: Q11718 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11718 / qualifier
 
xml-id: C17.S1.XMD7.m1edec

Latest revision as of 14:58, 2 January 2020

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DLMF:17.6.E11
No description defined

    Statements

    Defining formula
    1 - z 1 - b ⁢ ϕ 1 2 ⁡ ( q , a ⁢ q b ⁢ q ; q , z ) = ∑ n = 0 ∞ ( a ⁢ q ; q ) n ⁢ ( a ⁢ z ⁢ q / b ; q ) 2 ⁢ n ⁢ b n ( z ⁢ q , a ⁢ q / b ; q ) n - a ⁢ q ⁢ ∑ n = 0 ∞ ( a ⁢ q ; q ) n ⁢ ( a ⁢ z ⁢ q / b ; q ) 2 ⁢ n + 1 ⁢ ( b ⁢ q ) n ( z ⁢ q ; q ) n ⁢ ( a ⁢ q / b ; q ) n + 1 , 1 𝑧 1 𝑏 q-hypergeometric-rphis 2 1 𝑞 𝑎 𝑞 𝑏 𝑞 𝑞 𝑧 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 𝑞 𝑏 𝑞 2 𝑛 superscript 𝑏 𝑛 q-multiple-Pochhammer 𝑧 𝑞 𝑎 𝑞 𝑏 𝑞 𝑛 𝑎 𝑞 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑧 𝑞 𝑏 𝑞 2 𝑛 1 superscript 𝑏 𝑞 𝑛 q-Pochhammer-symbol 𝑧 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑏 𝑞 𝑛 1 {\displaystyle{\displaystyle\frac{1-z}{1-b}{{}_{2}\phi_{1}}\left({q,aq\atop bq% };q,z\right)=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}\left(azq/b;q\right% )_{2n}b^{n}}{\left(zq,aq/b;q\right)_{n}}-aq\sum_{n=0}^{\infty}\frac{\left(aq;q% \right)_{n}\left(azq/b;q\right)_{2n+1}(bq)^{n}}{\left(zq;q\right)_{n}\left(aq/% b;q\right)_{n+1}},}}
    0 references
    DLMF id
    DLMF:17.6.E11
    0 references
    constraint
    | z | < 1 , | b | < 1 formulae-sequence 𝑧 1 𝑏 1 {\displaystyle{\displaystyle|z|<1,|b|<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.m2agdec
    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.m2ajdec
    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.m6agdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1jdec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1ddec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1edec
    0 references
     
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