AI 中文总结
研究广义中心三项式系数，利用常数项方法和生成函数证明参数同余式，证实了孙志伟的几个猜想。
AI 中文摘要
对于任意\(n\in\mathbb{N}=\{0,1,2,\ldots\}\)以及\(b,c\in\mathbb{C}\)，\(n\)次广义中心三项式系数定义为\((b + x + cx^{-1})^n\)的洛朗展开式中的常数项。本文利用常数项方法和生成函数，证明了一个关于广义中心三项式系数的参数同余式，并应用其证实了孙志伟的几个猜想。
英文摘要
For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{C}$, the $n$-th generalized central trinomial coefficient is defined as the constant term in the Laurent expansion of $(b+x+cx^{-1})^n$. In this paper, utilizing the constant term method and generating functions, we prove several congruences concerning generalized central trinomial coefficients. First, we establish a parametric congruence modulo $p^2$, from which several conjectural congruences of Z.-W. Sun follow as special cases. We also prove a Catalan-type congruence modulo $p^2$ and a congruence involving $\binom{3k}{k}T_{3k}(6,1)$ modulo $p$, thereby confirming another two conjectural congruences of Z.-W. Sun.
Comments22 pages
