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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}("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.00476","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-12-28T21:51:52Z","cross_cats_sorted":[],"title_canon_sha256":"59daf57ee443f3ab777ff1344ad57c57c461c9eed661c951fc634d2581e18bdd","abstract_canon_sha256":"a2ff771e9d49c2fd0a4519006de7113d2f8723971db6facd9c3b27153ecb9d0d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:03.727484Z","signature_b64":"aYMQt4fEuTUf0Sj9d09sMkXiyxT6QPiPK0gCqZr0SaaBnu9Cm7EO3K49i/sj4fHz8knB5vWrRqVJ7SW8Zy+dDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77646d8df1e0035c789fef95a6158bf1c5dfc4936e44fd689b064c968f4d2b61","last_reissued_at":"2026-05-17T23:57:03.726863Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:03.726863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.00476","created_at":"2026-05-17T23:57:03.726987+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.00476v1","created_at":"2026-05-17T23:57:03.726987+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.00476","created_at":"2026-05-17T23:57:03.726987+00:00"},{"alias_kind":"pith_short_12","alias_value":"O5SG3DPR4ABV","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_16","alias_value":"O5SG3DPR4ABVY6E7","created_at":"2026-05-18T12:32:43.782077+00:00"},{"alias_kind":"pith_short_8","alias_value":"O5SG3DPR","created_at":"2026-05-18T12:32:43.782077+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H","json":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H.json","graph_json":"https://pith.science/api/pith-number/O5SG3DPR4ABVY6E756K2MFML6H/graph.json","events_json":"https://pith.science/api/pith-number/O5SG3DPR4ABVY6E756K2MFML6H/events.json","paper":"https://pith.science/paper/O5SG3DPR"},"agent_actions":{"view_html":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H","download_json":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H.json","view_paper":"https://pith.science/paper/O5SG3DPR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.00476&json=true","fetch_graph":"https://pith.science/api/pith-number/O5SG3DPR4ABVY6E756K2MFML6H/graph.json","fetch_events":"https://pith.science/api/pith-number/O5SG3DPR4ABVY6E756K2MFML6H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H/action/storage_attestation","attest_author":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H/action/author_attestation","sign_citation":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H/action/citation_signature","submit_replication":"https://pith.science/pith/O5SG3DPR4ABVY6E756K2MFML6H/action/replication_record"}},"created_at":"2026-05-17T23:57:03.726987+00:00","updated_at":"2026-05-17T23:57:03.726987+00:00"}