Applications of Differential-Difference Algebra in Discrete Calculus

Abstract

Discrete calculus deals with developing the concepts and techniques of differential and integral
calculus in a discrete setting, often using difference equations and discrete function spaces. This paper
explores how differential-difference algebra can provide an algebraic framework for advancing discrete
calculus. Differential-difference algebra studies algebraic structures equipped with both differential and
difference operators. These hybrid algebraic systems unify continuous and discrete analogues of
derivatives and shifts. This allows the development of general theorems and properties that cover both
settings. In particular, we construct differential-difference polynomial rings and fields over discrete
function spaces. We define discrete derivatives and shifts algebraically using these operators. We then
study integration, summation formulas, fundamental theorems, and discrete analogues of multivariate
calculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems in
differential-difference algebra that simultaneously yield results for both the continuous and discrete
cases. This provides new tools and insights for discrete calculus using modern algebraic techniques.
We also discuss applications of representing discrete calculus problems in differential-difference
algebras. This allows bringing to bear algebraic methods and software tools for their solution. Specific
examples are provided in areas such as numerical analysis of discrete dynamical systems defined
through difference equations. The paper aims to demonstrate the capabilities of differential-difference
algebra as a unifying framework for further developing the foundations and applications of discrete
calculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.