Congruences concerning generalized central trinomial coefficients via the constant term method

2026/07/09 by Chen Wang, Xia-Han Cui
Mathematics · #math.NT #math.CO

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Abstract

For any n∈ℕ=\0,1,2,…\ and b,c∈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 p², from which several conjectural congruences of Z.-W. Sun follow as special cases. We also prove a Catalan-type congruence modulo p² and a congruence involving \binom3kkT_3k(6,1) modulo p, thereby confirming another two conjectural congruences of Z.-W. Sun.

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