Formula:KLS:14.16:03

<< Formula:KLS:14.16:02

formula in Affine q-Krawtchouk

Formula:KLS:14.16:04 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle 0<pq<1}}}$

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
${\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}}}}$ : affine ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.16 of KLS.

URL links

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<< Formula:KLS:14.16:02

formula in Affine q-Krawtchouk

Formula:KLS:14.16:04 >>