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 lozenge(alpha) derivative defined as a linear combination of the Delta and the del derivatives, have promised improved approximation formulae for computational applications. This paper presents an equivalent definition of the lozenge(alpha) functions without reference to the existing Delta and del derivatives, examines the status of the lozenge(alpha) as a dynamic derivative and its properties relative to the Delta and del derivatives, and compares data obtained using the various dynamic derivatives as approximation formulae in computational experiments. A lozenge(alpha) integral case is investigated. (c) 2006 Elsevier Inc. All rights reserved.