Some Congruences Involving Fourth Powers of Generalized Central Trinomial Coefficients

TL;DR

This paper derives advanced congruences modulo prime powers for sums involving fourth powers of generalized central trinomial coefficients, revealing new relationships and specific formulas, especially for the case when parameters are set to particular values.

Contribution

It establishes novel congruences for sums of fourth powers of generalized central trinomial coefficients modulo p^3 and p^4, including explicit formulas for special cases.

Findings

Derived congruences modulo p^3 and p^4 for sums involving T_k(b,c)^4.

Provided explicit congruence formulas for the case b=c=1 involving Fermat quotients.

Extended understanding of the arithmetic properties of generalized trinomial coefficients.

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.