New Indefinite Summation Formulas and Some Applications

TL;DR

This paper introduces a new indefinite summation formula applicable to various difference equations and inequalities, providing a unified approach to find particular solutions and relating it to existing methods.

Contribution

It presents a novel indefinite summation formula for any function and demonstrates its applications to solve difference equations and inequalities, extending existing techniques.

Findings

New indefinite summation formula applicable to difference equations.

Unified method for solving linear difference equations and inequalities.

Relation established between new antidifference and existing results.

Abstract

In this paper, we introduce a novel indefinite summation $\sum_{t} f(t)$ (or antidifference $\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 $$ y(x+h)-\lambda y(x)=f(x),\quad h > 0,\quad \lambda \neq 0, $$ and also to solve a linear difference inequality of the form $$ y(x+h)-\lambda y(x) \geq 0,\quad h > 0,\quad \lambda \neq 0. $$ Furthermore, we apply the formula to determine a particular solution to a difference equations of the form $$ \Phi(E)y(t)=f(t), $$ and in solving a linear difference inequality of the form, $$ \Phi(E)y(t)\geq 0, $$ where $ \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.