Formula:KLS:14.21:20

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( γ t ; q ) ( t ; q ) \qHyperrphis 12 @ @ γ q α + 1 , γ t q - q α + 1 x t = n = 0 ( γ ; q ) n ( q α + 1 ; q ) n L n ( α ) ( x ; q ) t n q-Pochhammer-symbol 𝛾 𝑡 𝑞 q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 12 @ @ 𝛾 superscript 𝑞 𝛼 1 𝛾 𝑡 𝑞 superscript 𝑞 𝛼 1 𝑥 𝑡 superscript subscript 𝑛 0 q-Pochhammer-symbol 𝛾 𝑞 𝑛 q-Pochhammer-symbol superscript 𝑞 𝛼 1 𝑞 𝑛 q-Laguerre-polynomial-L 𝛼 𝑛 𝑥 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma t;q\right)_{% \infty}}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{2}@@{\gamma}{q^{\alpha+1}% ,\gamma t}{q}{-q^{\alpha+1}xt}{}=\sum_{n=0}^{\infty}\frac{\left(\gamma;q\right% )_{n}}{\left(q^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)t^{n}% }}}

Constraint(s)

γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


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
ϕ s r subscript subscript italic-ϕ 𝑠 𝑟 {\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}  : basic hypergeometric (or q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -hypergeometric) function : http://dlmf.nist.gov/17.4#E1
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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