Formula:KLS:14.15:04

From DRMF
Jump to navigation Jump to search


- ( 1 - q - x ) ⁢ K n ⁡ ( q - x ) = A n ⁢ K n + 1 ⁡ ( q - x ) - ( A n + C n ) ⁢ K n ⁡ ( q - x ) + C n ⁢ K n - 1 ⁡ ( q - x ) 1 superscript 𝑞 𝑥 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 subscript 𝐴 𝑛 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 subscript 𝐴 𝑛 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 subscript 𝐶 𝑛 q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle-\left(1-q^{-x}\right)K_{n}\!\left(q% ^{-x}\right)=A_{n}K_{n+1}\!\left(q^{-x}\right)-\left(A_{n}+C_{n}\right)K_{n}\!% \left(q^{-x}\right){}+C_{n}K_{n-1}\!\left(q^{-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

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
K n subscript 𝐾 𝑛 {\displaystyle{\displaystyle{\displaystyle K_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:qKrawtchouk

Bibliography

Equation in Section 14.15 of KLS.

URL links

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:14.15:04&oldid=5093"