Formula:KLS:14.14:19

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( q - x ⁢ t ; q ) x ⋅ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ q x - N , 0 ⁢ p ⁢ q ⁢ q ⁢ q - x ⁢ t = ∑ n = 0 N ( q - N ; q ) n ( p ⁢ q , q ; q ) n ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) ⁢ t n ⋅ q-Pochhammer-symbol superscript 𝑞 𝑥 𝑡 𝑞 𝑥 \qHyperrphis 21 @ @ superscript 𝑞 𝑥 𝑁 0 𝑝 𝑞 𝑞 superscript 𝑞 𝑥 𝑡 superscript subscript 𝑛 0 𝑁 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑛 q-Pochhammer-symbol 𝑝 𝑞 𝑞 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(q^{-x}t;q\right)_{x}\cdot% \qHyperrphis{2}{1}@@{q^{x-N},0}{pq}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(q^% {-N};q\right)_{n}}{\left(pq,q;q\right)_{n}}K^{\mathrm{qtm}}_{n}\!\left(q^{-x};% p,N;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
K n qtm subscript superscript 𝐾 qtm 𝑛 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}}}}  : quantum q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk

Bibliography

Equation in Section 14.14 of KLS.

URL links

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