{"paper":{"title":"Congruences for the Coefficients of the Powers of the Euler Product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Edward Y.S. Liu, Jack C.D. Zhao, Julia Q.D. Du","submitted_at":"2018-02-05T13:02:33Z","abstract_excerpt":"Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\\prod _{n=1}^{\\infty}(1-q^n)^k=\\sum_{n=0}^{\\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin $U$-operator, we determine the generating functions of $p_{8k}(2^{2\\alpha} n +\\frac{k(2^{2\\alpha}-1)}{3})$ $(1\\leq k\\leq 3)$ and $p_{3k} (3^{2\\beta}n+\\frac{k(3^{2\\beta}-1)}{8})$ $(1\\leq k\\leq 8)$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo $m$, we obtain infinite famil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01374","kind":"arxiv","version":2},"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"}