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

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 Apéry polynomials.

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