Formula:KLS:14.14:14

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∇ ⁡ [ w ⁢ ( x ; p , N ; q ) ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) ] ∇ ⁡ q - x = 1 1 - q ⁢ w ⁢ ( x ; p ⁢ q - 1 , N + 1 ; q ) ⁢ K n + 1 qtm ⁡ ( q - x ; p ⁢ q - 1 , N + 1 ; q ) ∇ 𝑤 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 ∇ superscript 𝑞 𝑥 1 1 𝑞 𝑤 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;p,N;q)K^{% \mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q\right)\right]}{\nabla q^{-x}}{}=\frac{1}% {1-q}w(x;pq^{-1},N+1;q)K^{\mathrm{qtm}}_{n+1}\!\left(q^{-x};pq^{-1},N+1;q% \right)}}}

Substitution(s)

w ⁢ ( x ; p , N ; q ) = ( q - N ; q ) x ( q , p - 1 ⁢ q - N ; q ) x ⁢ ( - p ) - x ⁢ q \binomial ⁢ x + 12 𝑤 𝑥 𝑝 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 superscript 𝑝 1 superscript 𝑞 𝑁 𝑞 𝑥 superscript 𝑝 𝑥 superscript 𝑞 \binomial 𝑥 12 {\displaystyle{\displaystyle{\displaystyle w(x;p,N;q)=\frac{\left(q^{-N};q% \right)_{x}}{\left(q,p^{-1}q^{-N};q\right)_{x}}(-p)^{-x}q^{\binomial{x+1}{2}}}}}


Proof

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Symbols List

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
( 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
( 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

Bibliography

Equation in Section 14.14 of KLS.

URL links

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