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

Abstract

The multi-variable Schmidt polynomials are defined by $$S_n^{(r)}(x_0,\dots ,x_n):=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n+k}{2k}\right)}^r\left(\genfrac{}{}{0ex}{}{2k}{k}\right)x_k.$$ We prove that, for any positive integers $m$, $n$, $r$, and $\epsilon =\pm 1$, all the coefficients in the polynomial $$\sum _{k=0}^{n-1}{\epsilon }^k(2k+1)S_k^{(r)}(x_0,\dots ,x_k{)}^m$$ are multiples of $n$. This generalizes a recent result of Pan on the divisibility of sums of Apéry polynomials.

6 pages

References in corpus (1)

• Some congruences involving powers of Delannoy polynomials