On the $p$-adic valuations of values of Legendre polynomials

TL;DR

This paper derives an explicit formula for the p-adic valuation of Legendre polynomial values at prime p, generalizes a previous conjecture, and resolves a problem posed by Cigler in 2017.

Contribution

It provides a new explicit formula for p-adic valuations of Legendre polynomials and addresses longstanding conjectures and open problems.

Findings

Derived an explicit p-adic valuation formula for Legendre polynomials.

Generalized an existing conjecture related to p-adic valuations.

Solved a problem proposed by Cigler in 2017.

Abstract

We prove an explicit formula for the $p$-adic valuation of the Legendre polynomials $P_n(x)$ evaluated at a prime $p$, and generalize an old conjecture of the third author. We also solve a problem proposed by Cigler in 2017.