Formula:KLS:14.21:19

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( t ; q ) ∞ ⋅ \qHyperrphis ⁢ 02 ⁢ @ ⁢ @ - q α + 1 , t ⁢ q - q α + 1 ⁢ x ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ q \binomial ⁢ n ⁢ 2 ( q α + 1 ; q ) n ⁢ L n ( α ) ⁡ ( x ; q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 02 @ @ superscript 𝑞 𝛼 1 𝑡 𝑞 superscript 𝑞 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot% \qHyperrphis{0}{2}@@{-}{q^{\alpha+1},t}{q}{-q^{\alpha+1}xt}=\sum_{n=0}^{\infty% }\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q^{\alpha+1};q\right)_{n}}L^{(\alpha% )}_{n}\!\left(x;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
L n ( α ) superscript subscript 𝐿 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:qLaguerre

Bibliography

Equation in Section 14.21 of KLS.

URL links

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