Generalized Legendre polynomials and related congruences modulo p²

2011/01/27 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #05A10 #05A19 #11A07 #11B39 #11B68 #11E25 #33C45 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1101.5386

openalex publication_date 2011/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any positive integer n and variables a and x we define the generalized Legendre polynomial P_n(a,x)=∑_k=0ⁿ\b ak\b-1-ak(\frac1-x2)^k. Let p be an odd prime. In the paper we prove many congruences modulo p² related to P_p-1(a,x). For example, we show that P_p-1(a,x)\e (-1)^_pP_p-1(a,-x)\mod p², where _p is the least nonnegative residue of a modulo p. We also generalize some congruences of Zhi-Wei Sun, and determine ∑_k=0^p-1\binom2kk\binom3kk54^-k and ∑_k=0^p-1\binom ak\binomb-ak\mod p², where [x] is the greatest integer function. Finally we pose some supercongruences modulo p² concerning binary quadratic forms.

Citations

• Supercongruences for a truncated hypergeometric series

Related

• Congruences concerning Legendre polynomials II
• Congruences concerning Legendre polynomials III
• New super congruences involving Bernoulli and Euler polynomials
• Jacobsthal sums, Legendre polynomials and binary quadratic forms
• A family of polynomials and related congruences and series