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

TL;DR

The paper derives explicit $p$-adic valuations $\nu_p(P_n(p))$ and $\nu_p(P_n(r))$ for Legendre polynomials, expressing them in terms of base-$p$ digit sums and binomial valuations, and proves a recurrence that yields $p$-regularity of the sequence $(\nu_p(P_n(p)))$ for every prime $p$. It provides a complete description for both odd primes and the case $p=2$, including a closed form $\nu_p(P_n(p)) = (2s_p(\lfloor n/2\rfloor) - s_p(n) + (n\bmod 2)p)/(p-1)$ and corresponding even/odd index formulas. The results imply $\nu_p(M_n(p)) = \nu_p(P_n(p))$ (Cigler’s conjecture) by relating $M_n(x)$ to $P_n(x)$ and extend known $3$-adic phenomena to general primes, offering recurrence relations that recover recent $p$-adic regularity findings (e.g., Shen 2023 for $p=3$). Altogether, the work advances understanding of $p$-adic valuations in Legendre polynomials and their combinatorial connections.

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.

Theorems & Definitions (17)

• Conjecture 1
• Conjecture 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 7
• Theorem 8
• Lemma 9
• proof