Supercongruences involving Lucas sequences

Abstract

For $A,B\in\mathbb Z$, the Lucas sequence $u_n(A,B)\ (n=0,1,2,\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\ldots).$ For any odd prime $p$ and positive integer $n$, we establish the new result $$\frac{u_{pn}(A,B) - (\frac{A^2-4B}p) u_n(A,B)}{pn} \in \mathbb Z_p,$$ where $(\frac{\cdot}p)$ is the Legendre symbol and $\mathbb Z_p$ is the ring of $p$-adic integers. Let $p$ be an odd prime and let $n$ be a positive integer. For any integer $m\not\equiv0\pmod p$, we show that $$\frac1{pn}\bigg(\sum_{k=0}^{pn-1} \frac{\binom{2k}k}{m^k} -\left(\frac{\Delta}p\right) \sum_{r=0}^{n-1}\frac{\binom{2r}r}{m^r}\bigg)\in\mathbb Z_p$$ and furthermore $$\frac1n\bigg(\sum_{k=0}^{pn-1} \frac{\binom{2k}k}{m^k} -\left(\frac{\Delta}p\right) \sum_{r=0}^{n-1}\frac{\binom{2r}r}{m^r}\bigg)\equiv \frac{\binom{2n-1}{n-1}}{m^{n-1}} u_{p-(\frac{\Delta}p)}(m-2,1) \pmod{p^2}$$ where $\Delta=m(m-4)$. We also pose some conjectures for further research.