Formula:KLS:14.15:03

<< Formula:KLS:14.15:02

formula in q-Krawtchouk

Formula:KLS:14.15:04 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle p>0}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

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
${\displaystyle{\displaystyle{\displaystyle K_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.15 of KLS.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:KLS:14.15:02

formula in q-Krawtchouk

Formula:KLS:14.15:04 >>