{"paper":{"title":"Higher-order identities for balancing numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Prasanta Kumar Ray, Takao Komatsu","submitted_at":"2016-08-21T12:15:46Z","abstract_excerpt":"Let $B_n$ be the $n$-th balancing number. In this paper, we give some explicit expressions of $\\sum_{l=0}^{2 r-3}(-1)^l\\binom{2 r-3}{l}\\sum_{j_1+\\cdots+j_r=n-2 l\\atop j_1,\\dots,j_r\\ge 1}B_{j_1}\\cdots B_{j_r}$ and $\\sum_{j_1+\\cdots+j_r=n\\atop j_1,\\dots,j_r\\ge 1}B_{j_1}\\cdots B_{j_r}$. We also consider the convolution identities with binomial coefficients: $$ \\sum_{k_1+\\cdots+k_r=n\\atop k_1,\\dots,k_r\\ge 1}\\binom{n}{k_1,\\dots,k_r}B_{k_1}\\cdots B_{k_r} $$ This type can be generalized, so that $B_n$ is a special case of the number $u_n$, where $u_n=a u_{n-1}+b u_{n-2}$ ($n\\ge 2$) with $u_0=0$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05925","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"}