Congruences concerning quadrinomial coefficients

TL;DR

This paper establishes new congruences modulo p^2 involving quadrinomial coefficients, extending known results for trinomial coefficients and contributing to the understanding of combinatorial congruences.

Contribution

It introduces novel congruences for quadrinomial coefficients modulo p^2, generalizing previous trinomial coefficient results.

Findings

Established congruences for quadrinomial coefficients mod p^2

Extended known trinomial coefficient congruences to quadrinomial case

Provided new tools for combinatorial number theory

Abstract

In this paper, we establish congruences (mod $p^2$) involving the quadrinomial coefficients $\dbinom{np-1}{p-1}_{3}$ and $\dbinom{np-1}{\frac{p-1}{2}}_{3}$. This is an analogue of congruences involving the trinomial coefficients $\dbinom{np-1}{p-1}_{2}$ and $\dbinom{np-1}{\frac{p-1}{2}}_{2}$ due to Elkhiri and Mihoubi.