{"paper":{"title":"Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers $h \\geq 2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Maxie D. Schmidt","submitted_at":"2017-02-05T06:10:07Z","abstract_excerpt":"We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\\binom{x+n}{n}$ and $\\binom{x}{n}$, over all $n \\geq 0$. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) $h \\geq 2$ and compare our results with existing known congruences for the binomial coefficients modulo primes $p$ and prime powers $p^k$. We also prove new exact formulas for these binomial coefficient cases from the expansions of the $h^{th}$ convergent functions to the infinite J-fraction series gener"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01374","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"}