Polynomials that generate a row of Pascal's triangle

Commun. Korean Math. Soc. 2002 Vol. 17, No. 3, 383-387 Printed September 1, 2002

Seon-Hong Kim Seoul National University

Abstract : Let $p$ be an odd prime, and let $f(x)$ be the interpolating polynomial associated with a table of data points $(j+1, \binom {p}{j})$ for $0 \leq j \leq p$. In this article, we find congruence identities modulo $p$ of $(p-1)! \, f(x)$, $(p-2)! \, f(x)$, and $(p-3)! \, f(x)$. Moreover we present some conjectures of these types.