Formula:KLS:14.16:04

{\displaystyle{\displaystyle{\displaystyle-(1-q^{-x})K^{\mathrm{Aff}}_{n}\!% \left(q^{-x}\right){}=(1-q^{n-N})(1-pq^{n+1})K^{\mathrm{Aff}}_{n+1}\!\left(q^{% -x}\right){}-\left[(1-q^{n-N})(1-pq^{n+1})-pq^{n-N}(1-q^{n})\right]K^{\mathrm{% Aff}}_{n}\!\left(q^{-x}\right){}-pq^{n-N}(1-q^{n})K^{\mathrm{Aff}}_{n-1}\!% \left(q^{-x}\right)}}}

Proof

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