Proof of some conjectural congruences involving Apéry and Apéry-like numbers
Sign Up to like & getrecommendations! Published in 2024 at "Proceedings of the Edinburgh Mathematical Society"
DOI: 10.1017/s0013091524000075
Abstract: Abstract In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then \begin{align*} &A_{2p}\equiv A_2-\frac{1648}3p^3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{2p-1}\equiv A_1+\frac{16p^3}3B_{p-3}\ ({\rm{mod}}\ p^4),\\ &A_{3p}\equiv A_3-36738p^3B_{p-3}\ ({\rm{mod}}\ p^4), \end{align*}… read more here.
Keywords: proof conjectural; congruences involving; involving like; conjectural congruences ... See more keywords
Proof of some conjectural congruences involving Apéry-like sequences
Sign Up to like & getrecommendations! Published in 2024 at "Journal of Difference Equations and Applications"
DOI: 10.1080/10236198.2024.2446458
Abstract: In this paper, we evaluate $$\begin{align*} \sum_{k=1}^{p-1}\frac1{k^3}\binom{x}k\binom{-1-x}k\quad \mbox{and}\quad \sum_{k=1}^{p-1}\frac1{k^2}\binom{x}k\binom{-1-x}k \end{align*}$$∑k=1p−11k3(xk)(−1−xk)and∑k=1p−11k2(xk)(−1−xk) modulo p and modulo $ p^2 $ p2 respectively, and by these we prove some conjectures of Z.-H. Sun involving $$\begin{align*} V_n(x)=\sum_{k=0}^n\binom{n}k\binom{n+k}k(-1)^k\binom{x}k\binom{-1-x}k \end{align*}$$Vn(x)=∑k=0n(nk)(n+kk)(−1)k(xk)(−1−xk) with n = p,… read more here.
Keywords: proof conjectural; congruences involving; align; conjectural congruences ... See more keywords