Matrix-induced quantum calculus in higher dimensions

In this work, we introduced a matrix-induced framework for quantum calculus on discrete forward orbits generated by affine contractions of the form $T(x)=a+Q(x-a)$ in ${\mathbb{R}}^N$, where the matrix $Q$ has a spectral radius less than one. The proposed approach extends the classical one-dimensional $q$-calculus to higher dimensions by replacing scalar dilations with matrix-driven dynamics. We defined matrix-induced directional derivatives and integrals on the associated forward orbit and established their fundamental properties, including integration by parts, and first and second fundamental theorems of calculus. As applications, we proved a Hermite–Hadamard-type inequality for convex functions under a segment-preserving condition on $Q$, derived a Poincaré-type inequality on the orbit, and obtained an existence–uniqueness result for a nonlinear first-order problem driven by the new derivative. In dimension one, the framework reduces to the standard $q$-calculus.