Abstract:

Given a sequence $x=\{x_n, \ n \in \mathbb{N}\}$ with integer values, or more generally with values in a ring of polynomials with integer coefficients, one can form the generalized binomial coefficients associated with $x$, ${\binom nm}_x=\prod_{l=1}^{m} \frac{x_{n-l+1}}{x_l}$. In this note we introduce several sequences that possess the following remarkable feature: the fractions $\binom nm_x$ are in fact polynomials with integer coefficients.