We introduce a novel definition of periodicity on arbitrary time scales, dependent on a strictly increasing and differentiable function. This removes the commonly used and restrictive assumption of a periodic time scale to define periodic functions. Our new definition furthermore allows for a wider class of functions to be studied using the theory of periodic systems. After providing crucial properties of these periodic functions, such as the translation invariance of integrals of periodic functions, we apply the concept of this new periodicity to linear dynamic equations. We provide necessary and sufficient conditions for a linear dynamic equation to have such a periodic solution and discuss its uniqueness.

In this paper, we introduce a novel formulation of dynamic Hardy-type inequalities on a time scale, motivated by a recently-established convexity approach in the Haar measure.The classical Hardy inequality is refined so that the classical Lebesgue-measure constant is replaced by the sharp constant $1$. We obtain time-scale analogues on finite intervals with best constants, and, for nonincreasing and nondecreasing functions, reversed inequalities with explicit weights described by incomplete $\beta$-functions. To establish our results, we employ two distinct time scales and apply the chain rule, together with the substitution rule, the derivative of inverse functions, and Fubini's theorem for delta integration. Our approach generalizes classical integral inequalities in the continuous setting, while yielding fundamentally new inequalities in the discrete setting. Furthermore, we explore the application of our results in the quantum case, demonstrating their broader relevance.