Diamond-alpha Polynomial Series on Time Scales

Abstract

The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more general notion of diamond-alpha derivative on time scales. We introduce a new notion of combined-polynomial series on a time scale, as a convex linear combination of delta and nabla generalized series. Main results are formulated for homogenous time scales. As an example, we compute diamond-alpha derivatives on time scales for delta and nabla exponential functions.

The Diamond Integral on Time Scales

“…For properties, results, and integral inequalities concerning the diamond-α integral, we refer the reader to [10,11,12,13] and references therein.…”

L'Hopital-Type Rules for Monotonicity with Application to Quantum Calculus

“…The theory of time scales was introduced in 1988 by Aulbach and Hilger in order to unify continuous and discrete-time theories (Aulbach and Hilger, 1990). It has found applications in several different fields that require simultaneous modeling of discrete and continuous data (Guseinov and Kaymakçalan, 2002;Wu and Debnath, 2009), and is now a subject of strong current research (see (Almeida and Torres, 2009;Bartosiewicz and Torres, 2008;Malinowska and Torres, 2009;Martins and Torres, 2009;Mozyrska and Torres, 2009;Sidi Ammi, Ferreira and Torres, 2008) and references therein). We claim that time scale theory is useful with respect to L'Hôpital-type rules for monotonicity, avoiding repetition of results in the continuous and discrete cases (Pinelis, 2007;Pinelis, 2008).…”

Diamond-Type Dirac Dynamic System in Mathematical Physics

“…Let t, t i ∈ T, p : T → R is a regressive function, p(t) ≡ p, and 1 + ν 2 (t)p 2 = 0 for all t ∈ T where ν(t) is the the backward graininess function. Then, for t ∈ T κ κ [24] sin α p (t,…”