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 $$\sum_{k=0}^{p-1} (2k+1)^{2a+1}\,\varepsilon^{k}\,\frac{T_k(b,c)^4}{d^{2k}},$$ where $a \in \left\lbrace 0,1\right\rbrace$, $\varepsilon \in \{1,-1\}$, 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.