Formula:KLS:14.17:04

<< Formula:KLS:14.17:02

formula in Dual q-Krawtchouk

Formula:KLS:14.17:05 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle c<0}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle K_{n}}}}$ : dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.17 of KLS.

URL links

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<< Formula:KLS:14.17:02

formula in Dual q-Krawtchouk

Formula:KLS:14.17:05 >>