Congruences concerning generalized central trinomial coefficients

TL;DR

This paper investigates congruences of generalized central trinomial coefficients modulo prime squares, providing explicit sums and confirming conjectures related to these coefficients.

Contribution

It derives new congruences for sums involving generalized central trinomial coefficients modulo prime squares and verifies existing conjectural congruences.

Findings

Explicit formulas for sums of squared coefficients modulo p^2

Confirmation of conjectural congruences by Sun

Extension of known results to generalized coefficients

Abstract

For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summation $\sum_{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400].