Formula:KLS:14.17:14

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K n ⁡ ( λ ⁢ ( x + 1 ) ; c , N | q ) - K n ⁡ ( λ ⁢ ( x ) ; c , N | q ) = q - n - x ⁢ ( 1 - q n ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N + 1 ) 1 - q - N ⁢ K n - 1 ⁡ ( λ ⁢ ( x ) ; c , N - 1 | q ) dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 1 𝑐 𝑁 𝑞 dual-q-Krawtchouk-polynomial-K 𝑛 𝜆 𝑥 𝑐 𝑁 𝑞 superscript 𝑞 𝑛 𝑥 1 superscript 𝑞 𝑛 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 1 superscript 𝑞 𝑁 dual-q-Krawtchouk-polynomial-K 𝑛 1 𝜆 𝑥 𝑐 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(\lambda(x+1);c,N|q% \right)-K_{n}\!\left(\lambda(x);c,N|q\right){}=\frac{q^{-n-x}(1-q^{n})(1-cq^{2% x-N+1})}{1-q^{-N}}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}}}} &
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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