Congruences concerning generalized central trinomial coefficients via the constant term method

Abstract

For any $n\in \mathbb{N}=\{0,1,2,\dots \}$ and $b,c\in \mathbb{C}$, the $n$-th generalized central trinomial coefficient is defined as the constant term in the Laurent expansion of $(b+x+c{x}^{-1}{)}^n$. 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 $\left(\genfrac{}{}{0ex}{}{3k}{k}\right)T_{3k}(6,1)$ modulo $p$, thereby confirming another two conjectural congruences of Z.-W. Sun.

22 pages