Formula:KLS:14.17:17

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[ w ( x ; c , N | q ) K n ( λ ( x ) ; c , N | q ) ] λ ( x ) = 1 ( 1 - q ) ( 1 - c q - N - 1 ) w ( x ; c , N + 1 | q ) K n + 1 ( λ ( x ) ; c , N + 1 | q ) 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 𝜆 𝑥 1 1 𝑞 1 𝑐 superscript 𝑞 𝑁 1 𝑤 𝑥 𝑐 𝑁 conditional 1 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;c,N|q)K_{n}\!% \left(\lambda(x);c,N|q\right)\right]}{\nabla\lambda(x)}{}=\frac{1}{(1-q)(1-cq^% {-N-1})}w(x;c,N+1|q)K_{n+1}\!\left(\lambda(x);c,N+1|q\right)}}}

Substitution(s)

λ ( n ) = q - n - p q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &

λ ( x ) = q - x + c q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}} &
w ( x ; c , N | q ) = ( q - N , c q - N ; q ) x ( q , c q ; q ) x c - x q 2 N x - x ( x - 1 ) 𝑤 𝑥 𝑐 conditional 𝑁 𝑞 q-Pochhammer-symbol superscript 𝑞 𝑁 𝑐 superscript 𝑞 𝑁 𝑞 𝑥 q-Pochhammer-symbol 𝑞 𝑐 𝑞 𝑞 𝑥 superscript 𝑐 𝑥 superscript 𝑞 2 𝑁 𝑥 𝑥 𝑥 1 {\displaystyle{\displaystyle{\displaystyle w(x;c,N|q)=\frac{\left(q^{-N},cq^{-% N};q\right)_{x}}{\left(q,cq;q\right)_{x}}c^{-x}q^{2Nx-x(x-1)}}}} &
λ ( x ) := q - x + c q x - N assign 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x):=q^{-x}+cq^{x-N}}}} &
λ ( n ) = q - n - p q n 𝜆 𝑛 superscript 𝑞 𝑛 𝑝 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle\lambda(n)=q^{-n}-pq^{n}}}} &

λ ( x ) = q - x + c q x - N 𝜆 𝑥 superscript 𝑞 𝑥 𝑐 superscript 𝑞 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=q^{-x}+cq^{x-N}}}}


Proof

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Symbols List

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

Bibliography

Equation in Section 14.17 of KLS.

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