Formula:KLS:14.14:05

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x ⁢ K ^ qtm n ⁡ ( x ) = K ^ qtm n + 1 ⁡ ( x ) + [ 1 - p - 1 ⁢ q - 2 ⁢ n - 1 ⁢ { ( 1 - q n - N ) + q ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) } ] ⁢ K ^ qtm n ⁡ ( x ) + p - 2 ⁢ q - 4 ⁢ n + 1 ⁢ ( 1 - q n ) ⁢ ( 1 - p ⁢ q n ) ⁢ ( 1 - q n - N - 1 ) ⁢ K ^ qtm n - 1 ⁡ ( x ) 𝑥 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 delimited-[] 1 superscript 𝑝 1 superscript 𝑞 2 𝑛 1 1 superscript 𝑞 𝑛 𝑁 𝑞 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 𝑥 𝑝 𝑁 𝑞 superscript 𝑝 2 superscript 𝑞 4 𝑛 1 1 superscript 𝑞 𝑛 1 𝑝 superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 𝑁 1 quantum-q-Krawtchouk-polynomial-monic-p 𝑛 1 𝑥 𝑝 𝑁 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{K}^{\mathrm{qtm}}}_{n}\!% \left(x\right)={\widehat{K}^{\mathrm{qtm}}}_{n+1}\!\left(x\right)+\left[1-p^{-% 1}q^{-2n-1}\left\{(1-q^{n-N})+q(1-q^{n})(1-pq^{n})\right\}\right]{\widehat{K}^% {\mathrm{qtm}}}_{n}\!\left(x\right){}+p^{-2}q^{-4n+1}(1-q^{n})(1-pq^{n})(1-q^{% n-N-1}){\widehat{K}^{\mathrm{qtm}}}_{n-1}\!\left(x\right)}}}

Proof

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Symbols List

K ^ qtm n subscript superscript ^ 𝐾 qtm 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{K}^{\mathrm{qtm}}}_{n}}}}  : monic quantum q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Krawtchouk polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqtmqKrawtchouk

Bibliography

Equation in Section 14.14 of KLS.

URL links

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