发表机构
School of Mathematics and Statistics Nanfang College; School of Mathematics and Statistics Huaiyin Normal University（南方学院数学与统计学院; 淮阴师范学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为两类Apéry型多项式建立加权双线性求和恒等式，结合端点值同余式得到模p^3和p^4的超同余式及特殊求值，并证实了Sun的一个猜想。
AI 中文摘要
我们为两类Apéry型多项式$g_n(x)$和$v_n(x)$建立了加权双线性求和恒等式。这些恒等式将加权和表示为连续端点值以及（必要时）低阶矩的表达式。对于$g_n(x)^2$，我们得到了权重为$(2n+1)^r$（$1\le r\le4$）的恒等式；对于$v_n(x)^2$，我们处理了三次和五次权重。将这些公式与端点值的同余式相结合，可以得到模$p^3$和$p^4$的超同余式，以及模$p^5$和$p^7$的特殊求值，其中$p$是大于$3$的素数。特别地，$$\sum_{n=0}^{p-1}(2n+1)^3v_n\\!\left(\frac52\right)^2 \equiv 6p^4-\frac{143}{3}p^6\pmod {p^7},$$ 证实了Sun猜想的一个同余式。证明使用了显式的二次 telescoping 恒等式和$p$-adic端点展开。
英文摘要
We establish weighted bilinear summation identities for two families of Apéry-like polynomials $g_n(x)$ and $v_n(x)$. The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For $g_n(x)^2$ we obtain identities with weights $(2n+1)^r$ for $1\le r\le4$; for $v_n(x)^2$ we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo $p^3$ and $p^4$, together with special evaluations modulo $p^5$ and $p^7$, where $p$ is a prime greater than $3$. In particular, $$\sum_{n=0}^{p-1}(2n+1)^3v_n\!\left(\frac52\right)^2 \equiv 6p^4-\frac{143}{3}p^6\pmod {p^7},$$ confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and $p$-adic endpoint expansions.
Comments29 pages
