Super-congruences involving trininomial coefficients

Abstract

The aim of this work is to establish congruences $\left(\mathrm{mod}{p}^2\right)$ involving the trinomial coefficients ${\left(\genfrac{}{}{0ex}{}{np-1}{p-1}\right)}_2$ and ${\left(\genfrac{}{}{0ex}{}{np-1}{\left(p-1\right)/2}\right)}_2$ 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 $\left(\genfrac{}{}{0ex}{}{np-1}{p-1}\right)$ and $\left(\genfrac{}{}{0ex}{}{np-1}{\left(p-1\right)/2}\right)$ and establish congruences link binomial coefficients and harmonic numbers.

References in corpus (3)

• Bernoulli Numbers, Wolstenholme's Theorem, and p^5 Variations of Lucas' Theorem
• Connection between ordinary multinomials, generalized Fibonacci numbers, partial Bell partition polynomials and convolution powers of discrete uniform distribution
• Some congruences for trinomial coefficients