Formula:KLS:14.14:02

<< Formula:KLS:14.14:01

formula in Quantum q-Krawtchouk

Formula:KLS:14.14:03 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{\left(pq;q\right)_{N-x}}{\left(q;q\right)_{x}\left(q;q\right)_{N-x}}(-1)^{N-x}q^{\binomial{x}{2}}K^{\mathrm{qtm}}_{m}\!\left(q^{-x};p,N;q\right)K^{\mathrm{qtm}}_{n}\!\left(q^{-x};p,N;q\right){}=\frac{(-1)^{n}p^{N}\left(q;q\right)_{N-n}\left(q,pq;q\right)_{n}}{\left(q,q;q\right)_{N}}q^{\binomial{N+1}{2}-\binomial{n+1}{2}+Nn}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle p>q^{-N}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{qtm}}_{n}}}}$ : quantum ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:qtmqKrawtchouk
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.14 of KLS.

URL links

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<< Formula:KLS:14.14:01

formula in Quantum q-Krawtchouk

Formula:KLS:14.14:03 >>