On the divisibility of sums involving powers of multi-variable Schmidt polynomials

Abstract

The multi-variable Schmidt polynomials are defined by [Formula: see text] We prove that, for any positive integers [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], all the coefficients in the polynomial [Formula: see text] are multiples of [Formula: see text]. This generalizes a recent result of Pan on the divisibility of sums of Apéry polynomials.