An algebraic analysis framework for quantum calculus

Abstract

An algebraic analysis framework for quantum calculus is proposed. The quantum derivative operator $D_{\tau ,\sigma }$ is based on two commuting bijections $\tau$ and $\sigma$ defined on an arbitrary set $M$ equipped with a tension structure determined by a single tension function $\theta$, i.e. a 1-dimensional case is analyzed here. The well known cases, i.e. $h$- and $q$-calculi together with their symmetric versions, can be obtained owing to special choice of mappings $\tau$ and $\sigma$.

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