Symmetric differentiation on time scales

While the functions in (1.1) do not have ordinary derivatives at t = 0, they have symmetric derivatives: f s (0) = g (0) = h (0) = 0. For a deeper understanding of the symmetric derivative and its properties, we refer the reader to the specialized monograph [8]. Here we note that the symmetric quotient (f (t+ h)− f (t− h)) /(2h) has, in general, better convergence properties than the ordinary difference quotient [6], leading naturally to the so-called h-symmetric quantum calculus [4]. A more recent theory is the general time-scale calculus. In 1988, Hilger introduced the calculus on time scales as a generalization of continuous and discrete time theories, obviating the need for separate proofs and highlighting the differences between them [1, 2]. Here we introduce the notion of symmetric derivative on time scales, initiating the corresponding theory and putting into context some of the recent results found in the literature. The article is organized as follows. In Section 2 we review the necessary concepts and we fix notations. The results are then given in Section 3, where we define the time scale symmetric derivative and derive some of its properties. Applications are found in the context of quantum calculus [4]. Finally, we show in Section 4 that the new symmetric derivative is a generalization of the diamond-α derivative [7], which brings new insights.