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Supercongruences for sums involving Domb numbers (2008.02647v2)
Published 6 Aug 2020 in math.NT and math.CO
Abstract: 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+p3E_{p-3} \pmod{p4}, \end{align*} which is regarded as a $p$-adic analogue of the following interesting formula for $1/\pi$ due to Rogers: \begin{align*} \sum_{k=0}{\infty}\frac{3k+1}{(-32)k}{\rm Domb}(k)=\frac{2}{\pi}. \end{align*} Here ${\rm Domb}(n)$ and $E_n$ are the famous Domb numbers and Euler numbers.