论文标题
关于中央二项式系数的总和三对一致
Three pairs of congruences concerning sums of central binomial coefficients
论文作者
论文摘要
最近，第一作者证明了Adamchuk在2006年提出的一致性：$ \ sum_ {k = 1}^{\ lfloor \ frac {2p} {2p} {3} \ rfloor} \ rfloor} \ binom {2k} {2k} {k} {k} {k} {k} {在本文中，我们提供了更多相同类型的一致性的示例（证明），$ \ sum_ {k = 1}^{\ lfloor \ frac {ap} {ap} {rfloor} \ rfloor} \ binom {binom {2k} \ pmod {r} $，$ a/r $是$（1/2,1）$的分数，$ x $是$ p $ - adic整数。关键成分是$ p $ -ADIC GAMMA功能$γ_P$和一类特殊的计算机发现的超几何身份。
Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where $p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions $Γ_p$ and a special class of computer-discovered hypergeometric identities.
