Formula:KLS:14.16:06

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x ⁢ K ^ n Aff ⁡ ( x ) = K ^ n + 1 Aff ⁡ ( x ) + [ 1 - { ( 1 - q n - N ) ⁢ ( 1 - p ⁢ q n + 1 ) - p ⁢ q n - N ⁢ ( 1 - q n ) } ] ⁢ K ^ n Aff ⁡ ( x ) - p ⁢ q n - N ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) ⁢ ( 1 - q n - N - 1 ) ⁢ K ^ n - 1 Aff ⁡ ( x ) 𝑥 affine-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 affine-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 affine-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝑁 1 affine-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x\widehat{K}^{\mathrm{Aff}}_{n}\!% \left(x\right)=\widehat{K}^{\mathrm{Aff}}_{n+1}\!\left(x\right)+\left[1-\left% \{(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^{n})\right\}\right]\widehat{K}^{\mathrm% {Aff}}_{n}\!\left(x\right){}-pq^{n-N}(1-q^{n})(1-pq^{n})(1-q^{n-N-1})\widehat{% K}^{\mathrm{Aff}}_{n-1}\!\left(x\right)}}}

Proof

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Symbols List

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

Bibliography

Equation in Section 14.16 of KLS.

URL links

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