Notes on the diamond-α dynamic derivative on time scales

Various dynamic derivative formulae have been proposed in the development of a time scales calculus, with the goal of unifying continuous and discrete analysis. Recent discussions of combined dynamic derivatives, in particular the α_? derivative defined as a linear combination of the Δ and the ∇ derivatives, have promised improved approximation formulae for computational applications. This paper presents an equivalent definition of the α_? functions without reference to the existing Δ and ∇ derivatives, examines the status of the α_? as a dynamic derivative and its properties relative to the Δ and ∇ derivatives, and compares data obtained using the various dynamic derivatives as approximation formulae in computational experiments. A α_? integral case is investigated.