{"paper":{"title":"A q-Analogue of Faulhaber's Formula for Sums of Powers","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2005-01-25T12:02:28Z","abstract_excerpt":"Let $$\n  S_{m,n}(q):=\\sum_{k=1}^{n}\\frac{1-q^{2k}}{1-q^2}\n  (\\frac{1-q^k}{1-q})^{m-1}q^{\\frac{m+1}{2}(n-k)}.\n  $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\\in\\mathbb{Z}[q]$ such that $$\n  S_{2m+1,n}(q) =\\sum_{k=0}^{m}(-1)^kP_{m,k}(q)\n  \\frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}}\n  {(1-q^2)(1-q)^{2m-3k}\\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ and solve a problem raised by Schlosser. We also show that there is a similar formula for the following $q$-analogue of alternating sums of powers:\n  $$\n  T_{m,n}(q):=\\sum_{k=1}^{n}(-1)^{n-k}\n  (\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501441","kind":"arxiv","version":3},"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"}