On the solutions of second order generalized difference equations

Abstract

In this article, the authors discuss ℓ2(ℓ) and c 0(ℓ) solutions of the second order generalized difference equation Δℓ2u(k) + f(k, u(k)) = 0, k ε [a,∞), a > 0 and we prove the condition for non existence of non-trivial solution where Δℓu(k) = u(k + ℓ) - u(k) for ℓ > 0. Further we present some formulae and examples to find the values of finite and infinite series in number theory as application of Δℓ. © 2012 Manuel et al.