{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:F6Z5KWWNGCKMCVPIOZURV6GMTX","short_pith_number":"pith:F6Z5KWWN","schema_version":"1.0","canonical_sha256":"2fb3d55acd3094c155e876691af8cc9de6608cff9f300873edbc18645a182dc7","source":{"kind":"arxiv","id":"1601.02192","version":1},"attestation_state":"computed","paper":{"title":"Some results associated with Bernoulli and Euler numbers with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"C.-P. Chen, R.B. Paris","submitted_at":"2016-01-10T09:37:28Z","abstract_excerpt":"In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\\mbox{sech} t$ and $\\coth t$.\n  For example, we prove that for $t > 0$ and $N\\in\\mathbb{N}:=\\{1, 2, \\ldots\\}$, \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t) \\] with \\[ R_N(t)=\\frac{(-1)^{N}2t^{2N}}{\\pi^{2N-1}}\\sum_{k=0}^{\\infty}\\frac{(-1)^{k}}{(k+\\frac{1}{2})^{2N-1}\\Big(t^2+\\pi^2(k+\\frac{1}{2})^2\\Big)}, \\] and \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+\\Theta(t, N)\\frac{E_{2N}}{(2N)!}t^{2N} \\] with a suitable $0 < \\Theta(t, N) < 1$. Here $E_n$ are the E"},"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":"1601.02192","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-10T09:37:28Z","cross_cats_sorted":[],"title_canon_sha256":"4ca7083c916480e91a453238980dee73941ed254e12861c0da90d9cf56e049c0","abstract_canon_sha256":"b5a2429b914c945f76ed761abce5cac461351ea9da6ad744e839c5a2f2d7f0c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:07.044643Z","signature_b64":"qk1LLfdh71JakDibbl2EvGd49Ya2+bus/TrKmaZHn1SyJIYW5E/vl6WgtEyiKDujlAX2NazZQ0W/wJwupN4/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fb3d55acd3094c155e876691af8cc9de6608cff9f300873edbc18645a182dc7","last_reissued_at":"2026-05-18T01:23:07.044080Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:07.044080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some results associated with Bernoulli and Euler numbers with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"C.-P. Chen, R.B. Paris","submitted_at":"2016-01-10T09:37:28Z","abstract_excerpt":"In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\\mbox{sech} t$ and $\\coth t$.\n  For example, we prove that for $t > 0$ and $N\\in\\mathbb{N}:=\\{1, 2, \\ldots\\}$, \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t) \\] with \\[ R_N(t)=\\frac{(-1)^{N}2t^{2N}}{\\pi^{2N-1}}\\sum_{k=0}^{\\infty}\\frac{(-1)^{k}}{(k+\\frac{1}{2})^{2N-1}\\Big(t^2+\\pi^2(k+\\frac{1}{2})^2\\Big)}, \\] and \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+\\Theta(t, N)\\frac{E_{2N}}{(2N)!}t^{2N} \\] with a suitable $0 < \\Theta(t, N) < 1$. Here $E_n$ are the E"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02192","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":"1601.02192","created_at":"2026-05-18T01:23:07.044189+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.02192v1","created_at":"2026-05-18T01:23:07.044189+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02192","created_at":"2026-05-18T01:23:07.044189+00:00"},{"alias_kind":"pith_short_12","alias_value":"F6Z5KWWNGCKM","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"F6Z5KWWNGCKMCVPI","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"F6Z5KWWN","created_at":"2026-05-18T12:30:15.759754+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/F6Z5KWWNGCKMCVPIOZURV6GMTX","json":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX.json","graph_json":"https://pith.science/api/pith-number/F6Z5KWWNGCKMCVPIOZURV6GMTX/graph.json","events_json":"https://pith.science/api/pith-number/F6Z5KWWNGCKMCVPIOZURV6GMTX/events.json","paper":"https://pith.science/paper/F6Z5KWWN"},"agent_actions":{"view_html":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX","download_json":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX.json","view_paper":"https://pith.science/paper/F6Z5KWWN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.02192&json=true","fetch_graph":"https://pith.science/api/pith-number/F6Z5KWWNGCKMCVPIOZURV6GMTX/graph.json","fetch_events":"https://pith.science/api/pith-number/F6Z5KWWNGCKMCVPIOZURV6GMTX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/action/storage_attestation","attest_author":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/action/author_attestation","sign_citation":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/action/citation_signature","submit_replication":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/action/replication_record"}},"created_at":"2026-05-18T01:23:07.044189+00:00","updated_at":"2026-05-18T01:23:07.044189+00:00"}