Formula:KLS:14.17:01

$\displaystyle {\displaystyle \dualqKrawtchouk{n}@{\lambda(x)}{c}{N}{q}=\qHyperrphis{3}{2}@@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q} }$

Constraint(s)

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

Substitution(s)

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

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

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

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
$\displaystyle {\displaystyle K_{n}}$  : dual $\displaystyle {\displaystyle q}$ -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:dualqKrawtchouk
$\displaystyle {\displaystyle {{}_{r}\phi_{s}}}$  : basic hypergeometric (or $\displaystyle {\displaystyle q}$ -hypergeometric) function : http://dlmf.nist.gov/17.4#E1