Formula:KLS:14.15:08: Difference between revisions

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Latest revision as of 08:38, 22 December 2019


x ⁢ K ^ n ⁡ ( x ) = K ^ n + 1 ⁡ ( x ) + [ 1 - ( A n + C n ) ] ⁢ K ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ K ^ n - 1 ⁡ ( x ) 𝑥 q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{K}}_{n}\!\left(x\right)=% {\widehat{K}}_{n+1}\!\left(x\right)+\left[1-(A_{n}+C_{n})\right]{\widehat{K}}_% {n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{K}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = - p ⁢ q 2 ⁢ n - N - 1 ⁢ ( 1 + p ⁢ q n + N ) ⁢ ( 1 - q n ) ( 1 + p ⁢ q 2 ⁢ n - 1 ) ⁢ ( 1 + p ⁢ q 2 ⁢ n ) subscript 𝐶 𝑛 𝑝 superscript 𝑞 2 𝑛 𝑁 1 1 𝑝 superscript 𝑞 𝑛 𝑁 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 1 𝑝 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}=-pq^{2n-N-1}\frac{(1+pq^{n+N}% )(1-q^{n})}{(1+pq^{2n-1})(1+pq^{2n})}}}} &
A n = ( 1 - q n - N ) ⁢ ( 1 + p ⁢ q n ) ( 1 + p ⁢ q 2 ⁢ n ) ⁢ ( 1 + p ⁢ q 2 ⁢ n + 1 ) subscript 𝐴 𝑛 1 superscript 𝑞 𝑛 𝑁 1 𝑝 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 𝑝 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n-N})(1+pq^{n})}{% (1+pq^{2n})(1+pq^{2n+1})}}}}


Proof

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Symbols List

& : logical and
K ^ n subscript ^ 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{K}}_{n}}}}  : monic q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqKrawtchouk

Bibliography

Equation in Section 14.15 of KLS.

URL links

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