{"paper":{"title":"Further Combinatorial Identities deriving from the $n$-th power of a $2 \\times 2$ matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"James Mc Laughlin, Nancy J. Wyshinski","submitted_at":"2018-12-28T21:51:52Z","abstract_excerpt":"In this paper we use a formula for the $n$-th power of a $2\\times2$ matrix $A$ (in terms of the entries in $A$) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if $m$ and $n$ are positive integers and $s \\in \\{0,1,2,\\dots,$ $\\lfloor (mn-1)/2 \\rfloor \\}$, then \\begin{multline*} \\sum_{i,j,k,t}2^{1+2t-mn+n} \\frac{(-1)^{nk+i(n+1)}}{1+\\delta_{(m-1)/2,\\,i+k}} \\binom{m-1-i}{i} \\binom{m-1-2i}{k}\\times\\\\ \\binom{n(m-1-2(i+k))}{2j}\\binom{j}{t-n(i+k)} \\binom{n-1-s+t}{s-t}\\\\ =\\binom{mn-1-s}{s}. \\end{multline*} 2) The generalized Fibonacci polynomial $f_{m}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.00476","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"}