论文标题
涉及DOMB数字的总和
Supercongruences for sums involving Domb numbers
论文作者
论文摘要
我们证明了涉及DOMB数量的总和，证明了Z.-W的四个猜想。太阳和Z.-H.太阳。例如，通过使用Chan和Zudilin引起的转换公式，我们表明，对于任何Prime $ p \ ge 5 $，\ begin {align*} \ sum_ {k = 0}^{p-1} \ frac} \ frac {3k+1} （-1）^{\ frac {p-1} {2}} p+p+p^3e_ {p-3} \ pmod {p^4}，\ end {align*}，该{align*}被视为$ 1/π$的以下有趣公式的$ p $ - adic-adic类似物\ sum_ {k = 0}^{\ infty} \ frac {3k+1} {（ - 32）^k} {\ rm domb}（k）= \ frac {2}π。 \ end {align*}这里$ {\ rm domb}（n）$和$ e_n $是著名的domb号码和欧拉号。
We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any prime $p\ge 5$, \begin{align*} \sum_{k=0}^{p-1}\frac{3k+1}{(-32)^k}{\rm Domb}(k)\equiv (-1)^{\frac{p-1}{2}}p+p^3E_{p-3} \pmod{p^4}, \end{align*} which is regarded as a $p$-adic analogue of the following interesting formula for $1/π$ due to Rogers: \begin{align*} \sum_{k=0}^{\infty}\frac{3k+1}{(-32)^k}{\rm Domb}(k)=\frac{2}π. \end{align*} Here ${\rm Domb}(n)$ and $E_n$ are the famous Domb numbers and Euler numbers.
