Formula:KLS:14.17:21

<< Formula:KLS:14.17:20

formula in Dual q-Krawtchouk

Formula:KLS:14.17:22 >>

${\displaystyle{\displaystyle{\displaystyle\left(cq^{-N}t;q\right)_{x}\cdot\left(q^{-N}t;q\right)_{N-x}=\sum_{n=0}^{N}\frac{\left(q^{-N};q\right)_{n}}{\left(q;q\right)_{n}}K_{n}\!\left(\lambda(x);c,N|q\right)t^{n}}}}$

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

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

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle K_{n}}}}$ : dual ${\displaystyle{\displaystyle{\displaystyle q}}}$-Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk

Bibliography

Equation in Section 14.17 of KLS.

URL links

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

<< Formula:KLS:14.17:20

formula in Dual q-Krawtchouk

Formula:KLS:14.17:22 >>