Combinatorial sums derived from properties of Legendre polynomials

TL;DR

This paper derives closed-form expressions for various combinatorial sums using properties of Legendre polynomials and related integrals.

Contribution

It introduces new closed-form formulas for combinatorial sums based on Legendre polynomial identities and integral results.

Findings

Derived closed-form formulas for combinatorial sums.

Connected combinatorial sums with Legendre polynomial properties.

Utilized integral results related to Legendre polynomials.

Abstract

From an identity connecting a combinatorial sum and Legendre polynomials, we derive closed forms for a number of combinatorial sums. Some of them are obtained via results about the integrals of functions associated with Legendre polynomials.