Formula:KLS:01.08:07

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( a ; q ) - n = 1 ( a ⁢ q - n ; q ) n = ( - q ⁢ a - 1 ) n ( q ⁢ a - 1 ; q ) n ⁢ q \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 1 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 superscript 𝑞 superscript 𝑎 1 𝑛 q-Pochhammer-symbol 𝑞 superscript 𝑎 1 𝑞 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{-n}=\frac{1}{\left% (aq^{-n};q\right)_{n}}=\frac{(-qa^{-1})^{n}}{\left(qa^{-1};q\right)_{n}}q^{% \binomial{n}{2}}}}}

Constraint(s)

n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}} &
a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}}


Substitution(s)

( a ⁢ q - n ; q ) n = ( a - 1 ⁢ q ; q ) n ⁢ ( - a ) n ⁢ q - n - \binomial ⁢ n ⁢ 2 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑎 1 𝑞 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 𝑛 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(aq^{-n};q\right)_{n}=\left(a^{% -1}q;q\right)_{n}(-a)^{n}q^{-n-\binomial{n}{2}}}}}


Proof

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Symbols List

& : logical and
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
( n k ) binomial 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1

Bibliography

Equation in Section 1.8 of KLS.

URL links

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