SUPERCONGRUENCES CONCERNING BISNOMIAL COEFFICIENTS

Abstract

We establish two supercongruences for ${\text{bi}}^s$nomial coefficients. We give a supercongruence similar to Jacobsthal’s binomial congruence, and as a consequence, we confirm the following conjecture for trinomial coefficients:

$${\left(\genfrac{}{}{0.0pt}{}{n{p}^r}{k{p}^r}\right)}_2\equiv {\left(\genfrac{}{}{0.0pt}{}{n{p}^{r-1}}{k{p}^{r-1}}\right)}_2(\mathrm{mod}{p}^{2r}),$$

where $n,k$ be nonnegative integers and $r\ge 1$ is an integer with $p>3$ is an odd prime number, which were posed by G.-S. Mao [On some congruences involving trinomial coefficients. Rocky Mountain J. Math. 50(5) (2020), 1759–1771]. We also generalize the Ljunggren congruence for binomial coefficients to ${\text{bi}}^s$nomial coefficients.