Formula:KLS:14.21:34

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x n = q - 1 2 ⁢ n ⁢ ( n + 2 ⁢ α + 1 ) ⁢ ( q α + 1 ; q ) n ⁢ ∑ k = 0 n ( q - n ; q ) k ( q α + 1 ; q ) k ⁢ q k ⁢ L k ( α ) ⁡ ( x ; q ) superscript 𝑥 𝑛 superscript 𝑞 1 2 𝑛 𝑛 2 𝛼 1 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 superscript subscript 𝑘 0 𝑛 q-Pochhammer-symbol superscript 𝑞 𝑛 𝑞 𝑘 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑘 superscript 𝑞 𝑘 q-Laguerre-polynomial-L 𝛼 𝑘 𝑥 𝑞 {\displaystyle{\displaystyle{\displaystyle x^{n}=q^{-\frac{1}{2}n(n+2\alpha+1)% }\left(q^{\alpha+1};q\right)_{n}\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}}% {\left(q^{\alpha+1};q\right)_{k}}q^{k}L^{(\alpha)}_{k}\!\left(x;q\right)}}}

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
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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