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arXiv · 2607.08300

Congruences concerning generalized central trinomial coefficients via the constant term method

Chen Wang·Xia-Han Cui
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Abstract

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.

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Chen Wang, Xia-Han Cui. 2026-07-09. Congruences concerning generalized central trinomial coefficients via the constant term method. https://arxiv.org/abs/2607.08300

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