DLMF:17.4.E3 (Q5352): Difference between revisions

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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.m1bdec
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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11714 / rank
 
Normal rank
Property / Symbols used: Q11714 / qualifier
 
Defining formula:

r 𝑟 {\displaystyle{\displaystyle r}}

r
Property / Symbols used: Q11714 / qualifier
 
xml-id: C17.S1.XMD5.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11717 / rank
 
Normal rank
Property / Symbols used: Q11717 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11717 / qualifier
 
xml-id: C17.S1.XMD6.m1adec
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.m1bdec

Latest revision as of 14:55, 2 January 2020

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DLMF:17.4.E3
No description defined

    Statements

    Defining formula
    ψ s r ⁡ ( a 1 , a 2 , … , a r b 1 , b 2 , … , b s ; q , z ) = ψ s r ⁡ ( a 1 , a 2 , … , a r ; b 1 , b 2 , … , b s ; q , z ) = ∑ n = - ∞ ∞ ( a 1 , a 2 , … , a r ; q ) n ⁢ ( - 1 ) ( s - r ) ⁢ n ⁢ q ( s - r ) ⁢ ( n 2 ) ⁢ z n ( b 1 , b 2 , … , b s ; q ) n = ∑ n = 0 ∞ ( a 1 , a 2 , … , a r ; q ) n ⁢ ( - 1 ) ( s - r ) ⁢ n ⁢ q ( s - r ) ⁢ ( n 2 ) ⁢ z n ( b 1 , b 2 , … , b s ; q ) n + ∑ n = 1 ∞ ( q / b 1 , q / b 2 , … , q / b s ; q ) n ( q / a 1 , q / a 2 , … , q / a r ; q ) n ⁢ ( b 1 ⁢ b 2 ⁢ ⋯ ⁢ b s a 1 ⁢ a 2 ⁢ ⋯ ⁢ a r ⁢ z ) n . q-hypergeometric-rpsis 𝑟 𝑠 subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 subscript 𝑏 1 subscript 𝑏 2 … subscript 𝑏 𝑠 𝑞 𝑧 q-hypergeometric-rpsis 𝑟 𝑠 subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 subscript 𝑏 1 subscript 𝑏 2 … subscript 𝑏 𝑠 𝑞 𝑧 superscript subscript 𝑛 q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 superscript 1 𝑠 𝑟 𝑛 superscript 𝑞 𝑠 𝑟 binomial 𝑛 2 superscript 𝑧 𝑛 q-multiple-Pochhammer subscript 𝑏 1 subscript 𝑏 2 … subscript 𝑏 𝑠 𝑞 𝑛 superscript subscript 𝑛 0 q-multiple-Pochhammer subscript 𝑎 1 subscript 𝑎 2 … subscript 𝑎 𝑟 𝑞 𝑛 superscript 1 𝑠 𝑟 𝑛 superscript 𝑞 𝑠 𝑟 binomial 𝑛 2 superscript 𝑧 𝑛 q-multiple-Pochhammer subscript 𝑏 1 subscript 𝑏 2 … subscript 𝑏 𝑠 𝑞 𝑛 superscript subscript 𝑛 1 q-multiple-Pochhammer 𝑞 subscript 𝑏 1 𝑞 subscript 𝑏 2 … 𝑞 subscript 𝑏 𝑠 𝑞 𝑛 q-multiple-Pochhammer 𝑞 subscript 𝑎 1 𝑞 subscript 𝑎 2 … 𝑞 subscript 𝑎 𝑟 𝑞 𝑛 superscript subscript 𝑏 1 subscript 𝑏 2 ⋯ subscript 𝑏 𝑠 subscript 𝑎 1 subscript 𝑎 2 ⋯ subscript 𝑎 𝑟 𝑧 𝑛 {\displaystyle{\displaystyle{{}_{r}\psi_{s}}\left({a_{1},a_{2},\dots,a_{r}% \atop b_{1},b_{2},\dots,b_{s}};q,z\right)={{}_{r}\psi_{s}}\left(a_{1},a_{2},% \dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)=\sum_{n=-\infty}^{\infty}\frac{% \left(a_{1},a_{2},\dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)\genfrac{(}{)}{% 0.0pt}{}{n}{2}}z^{n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n}}=\sum_{n=0}^{% \infty}\frac{\left(a_{1},a_{2},\dots,a_{r};q\right)_{n}(-1)^{(s-r)n}q^{(s-r)% \genfrac{(}{)}{0.0pt}{}{n}{2}}z^{n}}{\left(b_{1},b_{2},\dots,b_{s};q\right)_{n% }}+\sum_{n=1}^{\infty}\frac{\left(q/b_{1},q/b_{2},\dots,q/b_{s};q\right)_{n}}{% \left(q/a_{1},q/a_{2},\dots,q/a_{r};q\right)_{n}}\left(\frac{b_{1}b_{2}\cdots b% _{s}}{a_{1}a_{2}\cdots a_{r}z}\right)^{n}.}}
    0 references
    DLMF id
    DLMF:17.4.E3
    0 references
    Symbols used
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1aadec
    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.m6adec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1bdec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1adec
    0 references
    Q11714
    Defining formula
    r 𝑟 {\displaystyle{\displaystyle r}}
    xml-id
    C17.S1.XMD5.m1bdec
    0 references
    Q11717
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C17.S1.XMD6.m1adec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1bdec
    0 references
     
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