Repeated Sums and Binomial Coefficients

TL;DR

This paper introduces a novel definition of binomial coefficients as repeated sums of ones, providing new identities and simplifications for sums involving binomial coefficients and differences of sequences.

Contribution

It presents a new perspective on binomial coefficients through repeated sums and derives formulas for simplifying complex sums and differences.

Findings

New definition of binomial coefficients as repeated sums of ones

Simplification formulas for repeated sums and binomial-Harmonic sums

Relation between sequence differences and binomial coefficients

Abstract

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial coefficient identities will be shown in order to prove this definition. Using this new definition, we simplify some particular sums such as the repeated Harmonic sum and the repeated Binomial-Harmonic sum. We derive formulae for simplifying general repeated sums as well as a variant containing binomial coefficients. Additionally, we study the $m$-th difference of a sequence and show how sequences whose $m$-th difference is constant can be related to binomial coefficients.