Formula:KLS:14.11:20

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( a q t ; q ) \qHyperrphis 21 @ @ b q x - 1 , 0 b q q x t = n = 0 ( a q ; q ) n ( q ; q ) n P n ( x ; a , b ; q ) t n q-Pochhammer-symbol 𝑎 𝑞 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑏 𝑞 superscript 𝑥 1 0 𝑏 𝑞 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑞 𝑛 big-q-Laguerre-polynomial-P 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(aqt;q\right)_{\infty}\cdot% \qHyperrphis{2}{1}@@{bqx^{-1},0}{bq}{q}{xt}=\sum_{n=0}^{\infty}\frac{\left(aq;% q\right)_{n}}{\left(q;q\right)_{n}}P_{n}\!\left(x;a,b;q\right)t^{n}}}}

Proof

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Symbols List

( 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
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
P n subscript 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}}}}  : big q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqLaguerre

Bibliography

Equation in Section 14.11 of KLS.

URL links

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