论文标题
一致性的总和涉及三个二项式系数的产品
Congruences for sums involving products of three binomial coefficients
论文作者
论文摘要
令$ p> 3 $为素数，让$ a $为有理$ p $ - ad的整数，使用WZ方法，我们建立了$$ \ sum_ {k = 0}^{p-1}^{p-1} \ binom ak \ binom ak \ binom ak \ binom {-1-a-a} k \ binom {-1-a} k \ binom {k \ binom {k \ binom} {w（k）} {4^k}，$$其中$$ w（k）= 1，\ frac 1 {k+1}，\ frac 1 {（k+1）^2}，\ frac 1 {（k+1）^3} k，k^2，k^3，\ frac 1 {a+k}，\ frac 1 {a+k-1}。
Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\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.
