{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2005:42AURC7XG76SX42TEU6HRFO3WR","short_pith_number":"pith:42AURC7X","schema_version":"1.0","canonical_sha256":"e681488bf737fd2bf353253c7895dbb47f58f9f984969f8330f3299882adbf0a","source":{"kind":"arxiv","id":"math/0501441","version":3},"attestation_state":"computed","paper":{"title":"A q-Analogue of Faulhaber's Formula for Sums of Powers","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2005-01-25T12:02:28Z","abstract_excerpt":"Let $$\n  S_{m,n}(q):=\\sum_{k=1}^{n}\\frac{1-q^{2k}}{1-q^2}\n  (\\frac{1-q^k}{1-q})^{m-1}q^{\\frac{m+1}{2}(n-k)}.\n  $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\\in\\mathbb{Z}[q]$ such that $$\n  S_{2m+1,n}(q) =\\sum_{k=0}^{m}(-1)^kP_{m,k}(q)\n  \\frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}}\n  {(1-q^2)(1-q)^{2m-3k}\\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ and solve a problem raised by Schlosser. We also show that there is a similar formula for the following $q$-analogue of alternating sums of powers:\n  $$\n  T_{m,n}(q):=\\sum_{k=1}^{n}(-1)^{n-k}\n  (\\fra"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0501441","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2005-01-25T12:02:28Z","cross_cats_sorted":[],"title_canon_sha256":"a4e2af26354f65e41e5cbbadad1f858cb1f5da73a7ccded3d39de06ef8a52993","abstract_canon_sha256":"2005b2915b7eba6774e6537ab90c6389ce5e8350f1103a2bd332f10488a2abfd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:00.773394Z","signature_b64":"Z2O8l+5zd2QqHAVp8JP8bI+PkyzdtsecScS4pjsSU4zO3I+vc4xQm3gtf5GvEO1muiUSputd75f3M19FnUsxCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e681488bf737fd2bf353253c7895dbb47f58f9f984969f8330f3299882adbf0a","last_reissued_at":"2026-05-18T04:26:00.773016Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:00.773016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A q-Analogue of Faulhaber's Formula for Sums of Powers","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2005-01-25T12:02:28Z","abstract_excerpt":"Let $$\n  S_{m,n}(q):=\\sum_{k=1}^{n}\\frac{1-q^{2k}}{1-q^2}\n  (\\frac{1-q^k}{1-q})^{m-1}q^{\\frac{m+1}{2}(n-k)}.\n  $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\\in\\mathbb{Z}[q]$ such that $$\n  S_{2m+1,n}(q) =\\sum_{k=0}^{m}(-1)^kP_{m,k}(q)\n  \\frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}}\n  {(1-q^2)(1-q)^{2m-3k}\\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ and solve a problem raised by Schlosser. We also show that there is a similar formula for the following $q$-analogue of alternating sums of powers:\n  $$\n  T_{m,n}(q):=\\sum_{k=1}^{n}(-1)^{n-k}\n  (\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501441","kind":"arxiv","version":3},"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":"math/0501441","created_at":"2026-05-18T04:26:00.773073+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0501441v3","created_at":"2026-05-18T04:26:00.773073+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0501441","created_at":"2026-05-18T04:26:00.773073+00:00"},{"alias_kind":"pith_short_12","alias_value":"42AURC7XG76S","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_16","alias_value":"42AURC7XG76SX42T","created_at":"2026-05-18T12:25:52.687210+00:00"},{"alias_kind":"pith_short_8","alias_value":"42AURC7X","created_at":"2026-05-18T12:25:52.687210+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/42AURC7XG76SX42TEU6HRFO3WR","json":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR.json","graph_json":"https://pith.science/api/pith-number/42AURC7XG76SX42TEU6HRFO3WR/graph.json","events_json":"https://pith.science/api/pith-number/42AURC7XG76SX42TEU6HRFO3WR/events.json","paper":"https://pith.science/paper/42AURC7X"},"agent_actions":{"view_html":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR","download_json":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR.json","view_paper":"https://pith.science/paper/42AURC7X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0501441&json=true","fetch_graph":"https://pith.science/api/pith-number/42AURC7XG76SX42TEU6HRFO3WR/graph.json","fetch_events":"https://pith.science/api/pith-number/42AURC7XG76SX42TEU6HRFO3WR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR/action/storage_attestation","attest_author":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR/action/author_attestation","sign_citation":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR/action/citation_signature","submit_replication":"https://pith.science/pith/42AURC7XG76SX42TEU6HRFO3WR/action/replication_record"}},"created_at":"2026-05-18T04:26:00.773073+00:00","updated_at":"2026-05-18T04:26:00.773073+00:00"}