{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:F6Z5KWWNGCKMCVPIOZURV6GMTX","short_pith_number":"pith:F6Z5KWWN","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"},"canonical_sha256":"2fb3d55acd3094c155e876691af8cc9de6608cff9f300873edbc18645a182dc7","source":{"kind":"arxiv","id":"1601.02192","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.02192","created_at":"2026-05-18T01:23:07Z"},{"alias_kind":"arxiv_version","alias_value":"1601.02192v1","created_at":"2026-05-18T01:23:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02192","created_at":"2026-05-18T01:23:07Z"},{"alias_kind":"pith_short_12","alias_value":"F6Z5KWWNGCKM","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"F6Z5KWWNGCKMCVPI","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"F6Z5KWWN","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:F6Z5KWWNGCKMCVPIOZURV6GMTX","target":"record","payload":{"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"},"canonical_sha256":"2fb3d55acd3094c155e876691af8cc9de6608cff9f300873edbc18645a182dc7","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"},"source_kind":"arxiv","source_id":"1601.02192","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:23:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8SUzW/vP3NA5ju3XRmRujQMdq5Wfjoq4yDcXymaohjR1aWB8Q4h11vHRrCbFLBFe88gOgguB9RVNZbZ7bKj/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T03:11:24.580112Z"},"content_sha256":"57a9c1ffe75e2238674a3cf73f40da3fbe53291cdace8ad3706956c08725845f","schema_version":"1.0","event_id":"sha256:57a9c1ffe75e2238674a3cf73f40da3fbe53291cdace8ad3706956c08725845f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:F6Z5KWWNGCKMCVPIOZURV6GMTX","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:23:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ix2R5mrIFg9UThVMkShJVDLsGoamtCxTwvCGXMsCaalFK+vMG4HtRkVu+lW859g2EtevGbOwu258CdQOP08JBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T03:11:24.580469Z"},"content_sha256":"75a2df06c804afc1eb20dac993448e1010ecbe47d210588b7531b2f1c7bbe049","schema_version":"1.0","event_id":"sha256:75a2df06c804afc1eb20dac993448e1010ecbe47d210588b7531b2f1c7bbe049"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/bundle.json","state_url":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-25T03:11:24Z","links":{"resolver":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX","bundle":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/bundle.json","state":"https://pith.science/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F6Z5KWWNGCKMCVPIOZURV6GMTX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:F6Z5KWWNGCKMCVPIOZURV6GMTX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b5a2429b914c945f76ed761abce5cac461351ea9da6ad744e839c5a2f2d7f0c1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-10T09:37:28Z","title_canon_sha256":"4ca7083c916480e91a453238980dee73941ed254e12861c0da90d9cf56e049c0"},"schema_version":"1.0","source":{"id":"1601.02192","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.02192","created_at":"2026-05-18T01:23:07Z"},{"alias_kind":"arxiv_version","alias_value":"1601.02192v1","created_at":"2026-05-18T01:23:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02192","created_at":"2026-05-18T01:23:07Z"},{"alias_kind":"pith_short_12","alias_value":"F6Z5KWWNGCKM","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"F6Z5KWWNGCKMCVPI","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"F6Z5KWWN","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:75a2df06c804afc1eb20dac993448e1010ecbe47d210588b7531b2f1c7bbe049","target":"graph","created_at":"2026-05-18T01:23:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"C.-P. Chen, R.B. Paris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-10T09:37:28Z","title":"Some results associated with Bernoulli and Euler numbers with applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02192","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:57a9c1ffe75e2238674a3cf73f40da3fbe53291cdace8ad3706956c08725845f","target":"record","created_at":"2026-05-18T01:23:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b5a2429b914c945f76ed761abce5cac461351ea9da6ad744e839c5a2f2d7f0c1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-10T09:37:28Z","title_canon_sha256":"4ca7083c916480e91a453238980dee73941ed254e12861c0da90d9cf56e049c0"},"schema_version":"1.0","source":{"id":"1601.02192","kind":"arxiv","version":1}},"canonical_sha256":"2fb3d55acd3094c155e876691af8cc9de6608cff9f300873edbc18645a182dc7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2fb3d55acd3094c155e876691af8cc9de6608cff9f300873edbc18645a182dc7","first_computed_at":"2026-05-18T01:23:07.044080Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:07.044080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qk1LLfdh71JakDibbl2EvGd49Ya2+bus/TrKmaZHn1SyJIYW5E/vl6WgtEyiKDujlAX2NazZQ0W/wJwupN4/DA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:07.044643Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.02192","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57a9c1ffe75e2238674a3cf73f40da3fbe53291cdace8ad3706956c08725845f","sha256:75a2df06c804afc1eb20dac993448e1010ecbe47d210588b7531b2f1c7bbe049"],"state_sha256":"1e176e359df3ffc6affadf5292bf4cdb6a3b6b3a6e11d00fd9bcb65ca65543cc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ty7zPEFL2mh5gZVPdECvZXzN/vIOlWdGVN6CzvkpMz0PjkCj9Rs4rbMe4rC892cNf+1TSMUva6cNA3r1VpEOBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T03:11:24.582381Z","bundle_sha256":"f8ff01d71d036fe3f44caae47d62bcb6f21bd3247a6e74dc1268bb66390485f9"}}