Formula:KLS:01.08:12: Difference between revisions

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Latest revision as of 08:33, 22 December 2019


( a ; q ) n + k = ( a ; q ) n ( a q n ; q ) k q-Pochhammer-symbol 𝑎 𝑞 𝑛 𝑘 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 𝑘 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n+k}=\left(a;q% \right)_{n}\left(aq^{n};q\right)_{k}}}}

Constraint(s)

a 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\neq 0}}} &
0 < | q | < 1 0 𝑞 1 {\displaystyle{\displaystyle{\displaystyle 0<|q|<1}}}


Substitution(s)

( a ; q ) n = ( a ; q ) ( a q n ; q ) = ( a - 1 q 1 - n ; q ) n ( - a ) n q \binomial n 2 q-Pochhammer-symbol 𝑎 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 q-Pochhammer-symbol 𝑎 superscript 𝑞 𝑛 𝑞 q-Pochhammer-symbol superscript 𝑎 1 superscript 𝑞 1 𝑛 𝑞 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{n}=\frac{\left(a;q% \right)_{\infty}}{\left(aq^{n};q\right)_{\infty}}=\left(a^{-1}q^{1-n};q\right)% _{n}(-a)^{n}q^{\binomial{n}{2}}}}} &
( a ; q ) = k = 0 ( 1 - a q k ) = ( a ; q 2 ) ( a q ; q 2 ) q-Pochhammer-symbol 𝑎 𝑞 superscript subscript product 𝑘 0 1 𝑎 superscript 𝑞 𝑘 q-Pochhammer-symbol 𝑎 superscript 𝑞 2 q-Pochhammer-symbol 𝑎 𝑞 superscript 𝑞 2 {\displaystyle{\displaystyle{\displaystyle\left(a;q\right)_{\infty}=\prod_{k=0% }^{\infty}(1-aq^{k})=\left(a;q^{2}\right)_{\infty}\left(aq;q^{2}\right)_{% \infty}}}}


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
Π Π {\displaystyle{\displaystyle{\displaystyle\Pi}}}  : product : http://drmf.wmflabs.org/wiki/Definition:prod

Bibliography

Equation in Section 1.8 of KLS.

URL links

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