D-polynomials and Taylor formula in quantum calculus

Quantum calculus based on the right invertible divided difference operator $D_{\sigma }^{\tau }$ is proposed here in context of algebraic analysis \cite{DPR}. The linear operator $D_{\sigma }^{\tau }$, specified with the help of two fixed maps $\sigma ,\tau \text{\hspace{0.17em}⁣}:M\to M$, generalizes the quantum derivative operator used in $h$- or $q$-calculus \cite{kac}. In the domain of $D_{\sigma }^{\tau }$ there are special elements defined as $D_{\sigma }^{\tau }$-polynomials and the corresponding Taylor formula is proved.