Taylor polynomials for nabla dynamic equations on time scales

Abstract. We are concerned with the representation of polynomials for nabla dynamic equations on time scales. Once established, these polynomials will be used to derive Taylor’s Formula for dynamic functions. Several examples are given for special time scales such as Z, R, and qZ for q> 1. These polynomials will be related to those given for delta dynamic equations. 1. Preliminaries on Time Scales The following definitions can be found in Agarwal and Bohner [1], Atici and Gu-seinov [2], and Bohner and Peterson [3]; these are based on the notions first introduced by Hilger in his Ph.D. thesis [4]. A time scale T is any nonempty closed subset of R. It follows that the jump operators σ, ρ: T → T σ(t) = inf{s ∈ T: s> t} and ρ(t) = sup{s ∈ T: s < t} (supplemented by inf ∅: = supT and sup ∅: = inf T) are well defined. The point t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t)> t, respectively. If T has a right-scattered minimum m, define Tκ: = T − {m}; otherwise, set Tκ = T. If T has a left-scattered maximum M, define Tκ: = T−{M}; otherwise, set Tκ = T. The forward graininess is µ(t): = σ(t) − t. Similarly, the backward graininess is ν(t): = t − ρ(t). For f: T → R and t ∈ Tκ, the delta derivative [3] of f at t, denoted f∆(t), is the number (provided it exists) with the property that given any ε> 0, there is a neighborhood U of t such that |f(σ(t)) − f(s) − f∆(t)[σ(t) − s] | ≤ ε|σ(t) − s| for all s ∈ U. For T = R, we have f ∆ = f ′, the usual derivative, and for T = Z we have the forward difference operator, f∆(t) = f(t+ 1) − f(t). For f: T → R and t ∈ Tκ, the nabla derivative [2] of f at t, denoted f∇(t), is the number (provided it exists) with the property that given any ε> 0, there is