Some parametric congruences involving generalized central trinomial coefficients

2019/10/15 by Chen Wang, Zhi‐Wei Sun, Wang, Chen +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1910.06850

openalex publication_date 2019/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For n∈ℕ=\0,1,2,…\ and b,c∈ℤ, the nth generalized central trinomial coefficient T_n(b,c) is the coefficient of xⁿ in the expansion of (x²+bx+c)ⁿ. In particular, T_n=T_n(1,1) is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime p>3 ∑_k=0^p-1\frac\binom2kk12^kT_k≡((p)/(3))\frac3^p-1+34\pmodp² and ∑_k=0^p-1(T_kH_k)/(3^k)≡(3+((p)/(3)))/(2)-p(1+((p)/(3)))\pmodp², where (-) denotes the Legendre symbol and H_k:=∑_j=1^k1/j denotes the kth harmonic number. These confirm two conjectural congruences of the second author.

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