Differential and Integral Calculus of Sequence

TL;DR

This paper develops a calculus framework for sequences, defining derivatives, integrals, and functions, and extends classical identities and theorems into the sequence domain, including Fibonacci and factorial duals.

Contribution

It introduces a novel sequence calculus with new formalizations of functions, series, and identities, extending classical mathematical concepts to sequences.

Findings

Sequence calculus framework with derivatives and integrals.

Sequence versions of exponential, trigonometric, and hyperbolic functions.

New formalizations of Fibonacci sequence, factorial dual, and Bell number dual.

Abstract

We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function, hyperbolic function, trigonometric function, and also find sequence versions of the Maclaurin series for them. The sequence versions of exponential function involve divergent series including Grandi's series. By using this framework, we find a sequence version of the binomial theorem and Euler's identity. In addition, we design new formalisms of Fibonacci sequence and its generalizations. Last, we propose a sequence dual of factorial and Bell number, and find sequence dual of modular property of factorial concerning prime number (Wilson's theorem) and of Bell number concerning prime number.