[Paper Review] Generalized Legendre polynomials and related congruences modulo $p^2$

Zhi-Hong Sun|arXiv (Cornell University)|Jan 27, 2011

Advanced Mathematical Identities14 references3 citations

TL;DR

This paper introduces generalized Legendre polynomials $ P_n(a,x) $ to study congruences modulo $ p^2 $ for hypergeometric-type sums involving binomial coefficients. Using a recurrence relation and properties of $ p $-adic valuations, the authors prove that $ P_{p-1}(a,x) \equiv (-1)^{\langle a\rangle_p} P_{p-1}(a,-x) \pmod{p^2} $, which generalizes and unifies Rodriguez-Villegas and Sun's conjectures on supercongruences for sums like $ \sum_{k=0}^{p-1} \frac{\binom{2k}{k}^2}{16^k} \equiv \left(\frac{-1}{p}\right) \pmod{p^2} $. The key contribution is a new general congruence framework for such sums modulo $ p^2 $.

ABSTRACT

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that $P_{p-1}(a,x)\e (-1)^{_p}P_{p-1}(a,-x)\mod {p^2}$, where $_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k{54^{-k}}$ and $\sum_{k=0}^{p-1}\binom ak\binom{b-a}k\mod {p^2}$, where $[x]$ is the greatest integer function. Finally we pose some supercongruences modulo $p^2$ concerning binary quadratic forms.

Motivation & Objective

• To generalize Rodriguez-Villegas' and Sun's conjectural supercongruences modulo $ p^2 $ for hypergeometric sums involving binomial coefficients.
• To introduce and study the generalized Legendre polynomial $ P_n(a,x) $ as a unifying tool for analyzing such sums modulo $ p^2 $.
• To establish a recurrence relation for $ P_n(a,x) $ that enables modular reduction and congruence analysis at the $ p^2 $-level.
• To prove a general congruence for $ \sum_{k=0}^{p-1} \binom{a}{k}\binom{-1-a}{k} f(k) \equiv 0 \pmod{p^2} $ under certain conditions on $ f $, extending prior results.
• To pose and explore new conjectures on sums of the form $ \sum_{k=0}^{p-1} \binom{a}{k}\binom{b-a}{k} m^k \pmod{p^2} $.

Proposed method

• The authors define the generalized Legendre polynomial $ P_n(a,x) = \sum_{k=0}^n \binom{a}{k}\binom{a+k}{k} \left(\frac{x-1}{2}\right)^k $, which reduces to the classical Legendre polynomial when $ a = n $.
• They derive a key recurrence: $ (a+1)P_n(a+1,x) - (2a+1)xP_n(a,x) + aP_n(a-1,x) = -2(2a+1)\binom{a}{n}\binom{a+n}{n} \left(\frac{x-1}{2}\right)^{n+1} $, valid for $ a \not\equiv 0,-1 \pmod{p} $.
• By reducing this recurrence modulo $ p^3 $ and evaluating at $ n = p-1 $, they derive congruences for $ P_{p-1}(a,x) \pmod{p^2} $, especially focusing on the symmetry $ P_{p-1}(a,x) \equiv (-1)^{\langle a\rangle_p} P_{p-1}(a,-x) \pmod{p^2} $.
• They use $ p $-adic properties of binomial coefficients, including $ \binom{a}{k} \equiv \binom{\langle a\rangle_p}{k} \pmod{p} $, to reduce sums modulo $ p^2 $.
• They apply known results on Legendre symbols and representations of primes as $ x^2 + dy^2 $ to evaluate special cases, such as $ p = x^2 + 7y^2 $, in the context of $ \sum \binom{-1/4}{k}^2 64^k \pmod{p^2} $.
• They conjecture refined $ p^2 $-level congruences for sums like $ \sum \binom{-1/4}{k}^2 64^k \equiv (-1)^{\frac{x-1}{2}}(2x - \frac{p}{2x}) \pmod{p^2} $ when $ p = x^2 + 7y^2 $.

Experimental results

Research questions

• RQ1 How can generalized Legendre polynomials unify and extend known supercongruences modulo $ p^2 $ for hypergeometric sums?
• RQ2 What is the modular behavior of $ P_{p-1}(a,x) \pmod{p^2} $, and how does it relate to the $ p $-adic fractional part $ \langle a\rangle_p $?
• RQ3 Can the recurrence relation for $ P_n(a,x) $ be used to derive new congruences for sums of the form $ \sum_{k=0}^{p-1} \binom{a}{k}\binom{-1-a}{k} f(k) \pmod{p^2} $?
• RQ4 What are the precise $ p^2 $-level congruences for sums like $ \sum_{k=0}^{p-1} \binom{-1/4}{k}^2 64^k $, and how do they depend on the representation of $ p $ as $ x^2 + 7y^2 $?
• RQ5 Can the framework be extended to conjecture $ p^2 $-congruences for sums involving $ \binom{-1/6}{k}\binom{-5/6}{k} $ and other generalized binomial coefficients?

Key findings

• The main result is $ P_{p-1}(a,x) \equiv (-1)^{\langle a\rangle_p} P_{p-1}(a,-x) \pmod{p^2} $, which generalizes the symmetry of Legendre polynomials to the $ p^2 $-level.
• For $ a = -1/2, -1/3, -1/4, -1/6 $, this symmetry recovers Rodriguez-Villegas' conjectures modulo $ p^2 $, such as $ \sum_{k=0}^{p-1} \frac{\binom{2k}{k}^2}{16^k} \equiv \left(\frac{-1}{p}\right) \pmod{p^2} $.
• When $ \langle a\rangle_p $ is odd, the identity implies $ \sum_{k=0}^{p-1} \binom{a}{k}\binom{-1-a}{k} \frac{1}{2^k} \equiv 0 \pmod{p^2} $, generalizing earlier results.
• The paper proves $ \sum_{k=0}^{p-1} \binom{a}{k}\binom{-1-a}{k} \left( (-1)^{\langle a\rangle_p} f(k) - \sum_{m=0}^k \binom{k}{m} (-1)^m f(m) \right) \equiv 0 \pmod{p^2} $ for any $ p $-integer-valued function $ f $, a broad generalization.
• For $ f(k) = \binom{2k}{k}/4^k $, this yields $ \sum_{k=0}^{p-1} \binom{a}{k}\binom{-1-a}{k} \frac{\binom{2k}{k}}{4^k} \equiv 0 \pmod{p^2} $ when $ \langle a\rangle_p $ is odd.
• Conjecture 5.4 proposes $ p^2 $-level congruences for $ \sum \binom{-1/4}{k}^2 64^k \equiv (-1)^{\frac{x-1}{2}}(2x - \frac{p}{2x}) \pmod{p^2} $ when $ p = x^2 + 7y^2 \equiv 1 \pmod{4} $, and similar forms for other cases.