New Indefinite Summation Formulas and Some Applications

In this paper, we introduce a novel indefinite summation ${\sum }_{t}f(t)$ (or antidifference ${\mathrm{\Delta }}^{-1}f(t)$ ) formula for any given function $f$. We apply the indefinite summation formula to calculate a particular solution to a nonhomogeneous linear difference equation of the form $$y(x+h)-\lambda y(x)=f(x),h>0,\lambda \mathrm{\ne }0,$$ and also to solve a linear difference inequality of the form $$y(x+h)-\lambda y(x)\ge 0,h>0,\lambda \mathrm{\ne }0.$$ Furthermore, we apply the formula to determine a particular solution to a difference equations of the form $$\mathrm{\Phi }(E)y(t)=f(t),$$ and in solving a linear difference inequality of the form, $$\mathrm{\Phi }(E)y(t)\ge 0,$$ where $\mathrm{\Phi }(E)$ is some linear difference operator. We show how the antidifference of a function $f$ calculated with the current formula is related to the already existing result and establish the corresponding identity.