Formula:KLS:14.11:21

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( t ; q ) ∞ ⋅ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ 0 , 0 , x ⁢ a ⁢ q , b ⁢ q ⁢ q ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( q ; q ) n ⁢ P n ⁡ ( x ; a , b ; q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 32 @ @ 0 0 𝑥 𝑎 𝑞 𝑏 𝑞 𝑞 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑞 𝑞 𝑛 big-q-Laguerre-polynomial-P 𝑛 𝑥 𝑎 𝑏 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot% \qHyperrphis{3}{2}@@{0,0,x}{aq,bq}{q}{t}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{% \binomial{n}{2}}}{\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
( 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
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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