On Some Congruences Involving Generalized Central Trinomial Coefficients

Abstract

In this paper, we mainly prove some congruences involving generalized central trinomial coefficients

$$\begin{aligned} T_n(b,c):&=[x^n](x^2+bx+c)^n=[x^0](b+x+cx^{-1})^n\\&=\sum _{k=0}^{\lfloor n/2\rfloor }\left( {\begin{array}{c}n\\ 2k\end{array}}\right) \left( {\begin{array}{c}2k\\ k\end{array}}\right) b^{n-2k}c^k=\sum _{k=0}^{\lfloor n/2\rfloor }\left( {\begin{array}{c}n-k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) b^{n-2k}c^k, \end{aligned}$$

where $b, c\in \mathbb {Z}$. For example, let $p>3$ be a prime, b, c be integers and $p\not \mid d=b^2-4c$. Then,

$$\begin{aligned}&\sum _{k=0}^{p-1}(2k+1)^3\frac{T_k(b,c)^2}{d^k}\\&\quad \equiv {\left\{ \begin{array}{ll} -p^2\pmod {p^3} &{} \texttt{if}\ \textit{p}\mid \textit{c},\\ -p^2-\frac{2}{3}p^2\left( \frac{d}{p}\right) +p^2\left( \frac{7d}{6c}+\frac{d^2}{3c^2}\right) \left( \left( \frac{d}{p}\right) -1\right) \pmod {p^3} &{} \texttt{if} \ \textit{p}\not \mid \textit{c}, \end{array}\right. } \end{aligned}$$

where $\left( \frac{\cdot }{p}\right)$ denotes the Legendre symbol.