Formula:KLS:14.17:01

<< Affine q-Krawtchouk

formula in Dual q-Krawtchouk

Formula:KLS:14.17:02 >>

${\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x);c,N|q\right)=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}$

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

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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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.17 of KLS.

URL links

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<< Affine q-Krawtchouk

formula in Dual q-Krawtchouk

Formula:KLS:14.17:02 >>