Congruences for sums involving products of three binomial coefficients

TL;DR

This paper establishes new congruences modulo p^3 for sums involving products of three binomial coefficients and rational p-adic integers, using the WZ method, and applies these results to solve several conjectures.

Contribution

The paper introduces novel p-adic congruences for complex binomial sum expressions, extending previous conjectures and employing the WZ method for proof.

Findings

Derived congruences modulo p^3 for various binomial sums

Solved multiple conjectures related to binomial coefficient sums

Enhanced understanding of p-adic properties of binomial sums

Abstract

Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k\frac {w(k)}{4^k},$$ where $$w(k)=1,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3},\frac 1{2k-1},\frac 1{k+2}, \frac 1{k+3}, k,k^2,k^3,\frac 1{a+k},\frac 1{a+k-1}.$$ As consequences, taking $a=-\frac 12,-\frac 13,-\frac 14,-\frac 16$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.