Congruences concerning generalized central trinomial coefficients

2020/12/08 by Chen, Jia-Yu, Wang, Chen
#05A10 #11A07 #11B65 #11B75 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.04523

Abstract

For any n∈ℕ=\0,1,2,…\ and b,c∈ℤ, the generalized central trinomial coefficient T_n(b,c) denotes the coefficient of xⁿ in the expansion of (x²+bx+c)ⁿ. Let p be an odd prime. In this paper, we determine the summation ∑_k=0^p-1T_k(b,c)²/m^k modulo p² for integers m with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400].

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