A Note on the Difference of Powers and Falling Powers

Taoufik Sabar,Raúl E Curto

doi:10.1155/2021/5593780

Abstract

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.

Highlights

Identities involving binomial coefficients and combinatorial summations have several uses in mathematical physics and engineering and have appeared in several mathematical branches, such as combinatorics, probability theory, number theory, and graph theory. ese identities can be deduced by many techniques, switching the order of summations for double sums and mathematical induction, forcing telescoping, and sometimes, it is possible to interpret an expression as counting some quantity or computing the probability of some events
We use our main identity to convert from the difference of ordinary powers and falling powers to binomial coefficient. en, we obtain some other sums as a consequence of our main identity, and one of them is a result that was already proven in [4]
Our work differs from previous studies on combinatorial sums because it uses a new and simple mathematical induction principle rather than the ordinary complex methods mentioned above

Summary

Taoufik Sabar

Received 21 January 2021; Revised 4 March 2021; Accepted 19 March 2021; Published 5 April 2021. Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. Ey can be established by many techniques, from generating functions to special series. Using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. Is way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. E findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics

Introduction

Main Results

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