Proof of some conjectural congruences involving Domb numbers and binary quadratic forms

نویسندگان

چکیده

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then have $$ \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv4x^2-2p-\frac{p^2}{4x^2}\pmod{p^3}, if $p\equiv2\pmod3$, \sum_{k=0}^{p-1}\frac{D_k}{4^k}\equiv-2\sum_{k=0}^{p-1}\frac{D_k}{16^k}\equiv\frac{p^2}2\binom{\frac{p-1}2}{\frac{p-5}6}^{-2} \pmod{p^3}, where $D_n=\sum_{k=0}^n\binom{n}k^2\binom{2k}k\binom{2n-2k}{n-k}$ stands for $n$th Domb number.

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ژورنال

عنوان ژورنال: Journal of Mathematical Analysis and Applications

سال: 2022

ISSN: ['0022-247X', '1096-0813']

DOI: https://doi.org/10.1016/j.jmaa.2022.126493