Formula:KLS:14.17:10

<< Formula:KLS:14.17:09

formula in Dual q-Krawtchouk

Formula:KLS:14.17:14 >>

${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle D(x)=cq^{2x-2N-1}\frac{(1-q^{x})(1-cq^{x})}{(1-cq^{2x-N-1})(1-cq^{2x-N})}}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(1-q^{x-N})(1-cq^{x-N})}{(1-cq^{2x-N})(1-cq^{2x-N+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle y(x)=K_{n}\!\left(\lambda(x);c,N|q\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}}$ &

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

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

Bibliography

Equation in Section 14.17 of KLS.

URL links

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<< Formula:KLS:14.17:09

formula in Dual q-Krawtchouk

Formula:KLS:14.17:14 >>