Formula:KLS:14.17:09

<< Formula:KLS:14.17:08

formula in Dual q-Krawtchouk

Formula:KLS:14.17:10 >>

${\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,N|q\right)=\frac{1}{\left(q^{-N};q\right)_{n}}{\widehat{K}}_{n}\!\left(\lambda(x)\right)}}}$

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

Bibliography

Equation in Section 14.17 of KLS.

URL links

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<< Formula:KLS:14.17:08

formula in Dual q-Krawtchouk

Formula:KLS:14.17:10 >>