Formula:KLS:14.16:13

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Δ ⁢ K n Aff ⁡ ( q - x ; p , N ; q ) Δ ⁢ q - x = q - n + 1 ⁢ ( 1 - q n ) ( 1 - q ) ⁢ ( 1 - p ⁢ q ) ⁢ ( 1 - q - N ) ⁢ K n - 1 Aff ⁡ ( q - x ; p ⁢ q , N - 1 ; q ) Δ affine-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 Δ superscript 𝑞 𝑥 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 1 𝑝 𝑞 1 superscript 𝑞 𝑁 affine-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑞 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{\Delta K^{\mathrm{Aff}}_{n}\!% \left(q^{-x};p,N;q\right)}{\Delta q^{-x}}{}=\frac{q^{-n+1}(1-q^{n})}{(1-q)(1-% pq)(1-q^{-N})}K^{\mathrm{Aff}}_{n-1}\!\left(q^{-x};pq,N-1;q\right)}}}

Proof

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Symbols List

K n Aff subscript superscript 𝐾 Aff 𝑛 {\displaystyle{\displaystyle{\displaystyle K^{\mathrm{Aff}}_{n}}}}  : affine q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:AffqKrawtchouk

Bibliography

Equation in Section 14.16 of KLS.

URL links

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