Some parametric congruences involving generalized central trinomial coefficients

TL;DR

This paper establishes parametric congruences involving generalized central trinomial coefficients and confirms two conjectural congruences related to these coefficients and harmonic numbers modulo prime squares.

Contribution

It introduces new parametric congruences for generalized central trinomial coefficients and proves two conjectures involving sums with binomial coefficients and harmonic numbers.

Findings

Proves a congruence involving binomial coefficients and trinomial coefficients modulo p^2.

Establishes a congruence involving harmonic numbers and trinomial coefficients modulo p^2.

Confirms two conjectural congruences related to these coefficients.

Abstract

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central trinomial coefficient. In this paper, we mainly establish some parametric congruences involving generalized central trinomial coefficients. As consequences, we prove that for any prime $p>3$ $$ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}}{12^k}T_k\equiv\left(\frac{p}{3}\right)\frac{3^{p-1}+3}{4}\pmod{p^2} $$ and $$ \sum_{k=0}^{p-1}\frac{T_kH_k}{3^k}\equiv\frac{3+\left(\frac{p}{3}\right)}{2}-p\left(1+\left(\frac{p}{3}\right)\right)\pmod{p^2}, $$ where $(-)$ denotes the Legendre symbol and $H_k:=\sum_{j=1}^k1/j$ denotes the $k$th harmonic number. These confirm two conjectural congruences of the second author.