Formula:KLS:14.16:14

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