Some Congruences Involving Fourth Powers of Generalized Central Trinomial Coefficients

Let $p\ge 5$ be a prime and let $b,c\in \mathbb{Z}$. Denote by $T_k(b,c)$ the generalized central trinomial coefficient, i.e., the coefficient of $x^k$ in $(x^2+bx+c{)}^k$. In this paper, we establish congruences modulo $p^3$ and $p^4$ for sums of the form $$\sum _{k=0}^{p-1}(2k+1{)}^{2a+1}{\epsilon }^k\frac{T_k(b,c{)}^4}{d^{2k}},$$ where $a\in \{0,1\}$, $\epsilon \in \{1,-1\}$, and $d=b^2-4c$ satisfies $p\nmid d$. In particular, for the special case $b=c=1$, we show that \begin{align*} \sum_{k=0}^{p-1}\left( 2k+1\right) ^{3} \frac{T_{k}^4}{9^k}\equiv -\frac{3p}{4}+\frac{3p^2}{4}\left( \frac{q_p(3)}{4}-1\right) \pmod{p^3}, \end{align*} where $T_k$ is the central trinomial coefficient and $q_p(a)$ is the Fermat quotient.