Congruences for sums involving products of three binomial coefficients

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 $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{a}{k}\right)\left(\genfrac{}{}{0ex}{}{-1-a}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\frac{w(k)}{4^k},$$ 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 $a=-\frac{1}{2},-\frac{1}{3},-\frac{1}{4},-\frac{1}{6}$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.

arXiv commentadd Section 11 and more conjectures