Some parametric congruences involving generalized central trinomial coefficients

Abstract

For $n\in \mathbb{N}=\{0,1,2,\dots \}$ and $b,c\in \mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c{)}^n$. 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 $p>3$ $$\sum _{k=0}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{12^k}{T}_k\equiv \left(\frac{p}{3}\right)\frac{3^{p-1}+3}{4}(\mathrm{mod}{p}^2)$$ and $$\sum _{k=0}^{p-1}\frac{T_k{H}_k}{3^k}\equiv \frac{3+\left(\frac{p}{3}\right)}{2}-p\left(1+\left(\frac{p}{3}\right)\right)(\mathrm{mod}{p}^2),$$ where $(-)$ denotes the Legendre symbol and $H_k:={\sum }_{j=1}^{k}1/j$ denotes the $k$th harmonic number. These confirm two conjectural congruences of the second author.

arXiv comment14 pages