Super-congruences involving trininomial coefficients

TL;DR

This paper establishes new super-congruences modulo p^2 involving trinomial coefficients from powers of 1+x+x^2, extending known binomial coefficient congruences and linking them to harmonic numbers.

Contribution

It introduces novel super-congruences for trinomial coefficients modulo p^2, extending existing binomial coefficient congruences and connecting them with harmonic numbers.

Findings

Extended known binomial congruences to trinomial coefficients

Established congruences linking trinomial coefficients and harmonic numbers

Provided new super-congruences modulo p^2 for trinomial coefficients

Abstract

The aim of this work is to establish congruences $\left( \operatorname{mod}p^{2}\right) $ involving the trinomial coefficients $\binom{np-1}{p-1}_{2}$ and $\binom{np-1}{\left( p-1\right)/2}_{2}$ arising from the expansion of the powers of the polynomial $1+x+x^{2}.$ In main results we extend some known congruences involving the binomial coefficients $\binom{np-1}{p-1}$ and $\binom{np-1}{\left( p-1\right) /2}$ and establish congruences link binomial coefficients and harmonic numbers.