{"paper":{"title":"On the divisibility of sums involving powers of multi-variable Schmidt polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Qi-Fei Chen, Victor J. W. Guo","submitted_at":"2014-12-18T07:07:22Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5734","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}