Some Explicit Formulas for Sums Involving the Binomial Coefficients with the Falling Factorial

TL;DR

This paper introduces explicit formulas for sums involving binomial coefficients, Fibonacci, and Lucas numbers using finite differences and falling factorials, providing new closed-form expressions.

Contribution

It presents novel closed-form formulas for combinatorial sums involving binomial coefficients and special number sequences using finite difference techniques.

Findings

Derived explicit formulas for sums with binomial coefficients and Fibonacci numbers

Connected combinatorial sums to falling factorial expressions

Enhanced evaluation methods for related sums in combinatorics

Abstract

Spivey presented a new approach to evaluate combinatorial sums by using finite differences. We present some closed forms for sums involving the binomial coefficients, Fibonacci and Lucas numbers in terms of the falling factorial.