Formula:KLS:14.14:13

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( 1 - q x - N - 1 ) ⁢ K n qtm ⁡ ( q - x ; p , N ; q ) + q - x ⁢ ( 1 - q x ) ⁢ ( p - q x - N - 1 ) ⁢ K n qtm ⁡ ( q - x + 1 ; p , N ; q ) = ( 1 - q - N - 1 ) ⁢ K n + 1 qtm ⁡ ( q - x ; p ⁢ q - 1 , N + 1 ; q ) 1 superscript 𝑞 𝑥 𝑁 1 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 𝑝 𝑁 𝑞 superscript 𝑞 𝑥 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 𝑥 𝑁 1 quantum-q-Krawtchouk-polynomial-K 𝑛 superscript 𝑞 𝑥 1 𝑝 𝑁 𝑞 1 superscript 𝑞 𝑁 1 quantum-q-Krawtchouk-polynomial-K 𝑛 1 superscript 𝑞 𝑥 𝑝 superscript 𝑞 1 𝑁 1 𝑞 {\displaystyle{\displaystyle{\displaystyle(1-q^{x-N-1})K^{\mathrm{qtm}}_{n}\!% \left(q^{-x};p,N;q\right){}+q^{-x}(1-q^{x})(p-q^{x-N-1})K^{\mathrm{qtm}}_{n}\!% \left(q^{-x+1};p,N;q\right){}=(1-q^{-N-1})K^{\mathrm{qtm}}_{n+1}\!\left(q^{-x}% ;pq^{-1},N+1;q\right)}}}

Proof

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Symbols List

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

Bibliography

Equation in Section 14.14 of KLS.

URL links

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