Formula:KLS:14.17:06

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- ( 1 - q - x ) ⁢ ( 1 - c ⁢ q x - N ) ⁢ K n ⁡ ( λ ⁢ ( x ) ) = ( 1 - q n - N ) ⁢ K n + 1 ⁡ ( λ ⁢ ( x ) ) - [ ( 1 - q n - N ) + c ⁢ q - N ⁢ ( 1 - q n ) ] ⁢ K n ⁡ ( λ ⁢ ( x ) ) + c ⁢ q - N ⁢ ( 1 - q n ) ⁢ K n - 1 ⁡ ( λ ⁢ ( x ) ) 1 superscript 𝑞 𝑥 1 𝑐 superscript 𝑞 𝑥 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 1 superscript 𝑞 𝑛 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 𝑞 delimited-[] 1 superscript 𝑞 𝑛 𝑁 𝑐 superscript 𝑞 𝑁 1 superscript 𝑞 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 𝑐 superscript 𝑞 𝑁 1 superscript 𝑞 𝑛 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-(1-q^{-x})(1-cq^{x-N})K_{n}\!\left(% \lambda(x)\right){}=(1-q^{n-N})K_{n+1}\!\left(\lambda(x)\right){}-\left[(1-q^{% n-N})+cq^{-N}(1-q^{n})\right]K_{n}\!\left(\lambda(x)\right){}+cq^{-N}(1-q^{n})% K_{n-1}\!\left(\lambda(x)\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}}}} &
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

Bibliography

Equation in Section 14.17 of KLS.

URL links

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