Some Congruences Involving Fourth Powers of Generalized Central Trinomial Coefficients
Yassine Otmani, Hacene Belbachir
TL;DR
This work studies $p$-adic congruences modulo $p^3$ and $p^4$ for sums involving the fourth power of generalized central trinomial coefficients $T_k(b,c)$. The authors develop a framework based on Sun's identity, binomial-sum manipulations, and $p$-adic analytic tools (Fermat quotients, finite polylogarithms) to compute high-order terms in the expansions, distinguishing cases $p\mid c$, $p\mid b$, and $p\nmid bc$. They obtain explicit congruences for sums of the form $\sum_{k=0}^{p-1} (2k+1)^{2a+1}\varepsilon^{k}\frac{T_k(b,c)^4}{d^{2k}}$ with $a\in\{0,1\}$ and $\varepsilon\in\{1,-1\}$, and present a concrete $b=c=1$ case involving the Fermat quotient $q_p(3)$. Overall, the paper extends known quadratic-distance congruences for $T_k(b,c)^2$ to fourth powers, enriching the number-theoretic understanding of trinomial coefficients and their p-adic properties.
Abstract
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.