Formula:KLS:14.17:16

<< Formula:KLS:14.17:15

formula in Dual q-Krawtchouk

Formula:KLS:14.17:17 >>

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

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

Bibliography

Equation in Section 14.17 of KLS.

URL links

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<< Formula:KLS:14.17:15

formula in Dual q-Krawtchouk

Formula:KLS:14.17:17 >>