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

TL;DR

This paper proves that certain sums involving powers of multi-variable Schmidt polynomials have coefficients divisible by a given integer, extending previous results on polynomial divisibility.

Contribution

It establishes a general divisibility property for sums of powers of multi-variable Schmidt polynomials, generalizing prior work on Apéry polynomials.

Findings

Coefficients in the specified sums are divisible by n.

The result generalizes previous divisibility results for Apéry polynomials.

The proof applies to all positive integers m, n, r, and ε=±1.

Abstract

The multi-variable Schmidt polynomials are defined by $$ S_n^{(r)}(x_0,\ldots,x_n):=\sum_{k=0}^n {n+k \choose 2k}^{r}{2k\choose k} x_k. $$ We prove that, for any positive integers $m$, $n$, $r$, and $\varepsilon=\pm 1$, all the coefficients in the polynomial $$ \sum_{k=0}^{n-1}\varepsilon^k(2k+1) S_k^{(r)}(x_0,\ldots,x_k)^m $$ are multiples of $n$. This generalizes a recent result of Pan on the divisibility of sums of Ap\'ery polynomials.