Congruences with Sums of powers of quadrinomial coefficients

Abstract

Quadrinomial coefficients \(\binom{n}{k}_{3}\) defined by
\[
(1+x+x^2+x^3)^n= \sum\limits_{k=0}^{3n}\binom{n}{k}_{3}x^k.
\]
Let \(p>3\) be a prime number and \(n,m\) be positive integers, we obtained the congruences modulo \(p^2\) with partial sums of powers of quadrinomial coefficients
\[\sum\limits_{0\leq 4k+i\leq p-1}\binom{np-1}{4k+i}_{3}^{m} \text{~and~} \sum\limits_{0\leq 4k+i\leq \frac{p-1}{2}}\binom{np-1}{4k+i}_{3}^{m}(0\leq i\leq 3).\]\\
We also studied the congruences modulo \(p^2\) with sums and alternating sums of powers of quadrinomial coefficients
\[\sum\limits_{k=0}^{p-1}\binom{np-1}{k}_{3}^{m}, \sum\limits_{k=0}^{\frac{p-1}{2}}\binom{np-1}{k}_{3}^{m}, \sum\limits_{k=0}^{p-1}(-1)^k\binom{np-1}{k}_{3}^{m} \text{~and~} \sum\limits_{k=0}^{\frac{p-1}{2}}(-1)^k\binom{np-1}{k}_{3}^{m}.\]