Discrete Bernoulli's formula and its applications arising from generalized difference operator

In this paper, the authors derive discrete Bernoulli's formula of the form, [Delta_ell^{-1}[u(k) v(k)] = sumlimits_{t=0}^infty (-1)^t u^{(t)}(k) v_{t+1}(k+tell)]for the real valued real variable functions $u(k)$ and $v(k)$ using the generalized differences, $u^{(t)}(k)=Delta_ell^t u(k)$ and its inverses, $v_{t}(k)=Delta_ell^{-t} v(k)$ and obtain several formulae on finite and infinite series as application of the discrete Bernoulli's formula in number theory. Suitable examples are provided to illustrate the main results.