{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:HKDG5NFWGNQP5CYW33Z4L3AGAG","short_pith_number":"pith:HKDG5NFW","canonical_record":{"source":{"id":"1812.10764","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-12-27T16:42:52Z","cross_cats_sorted":[],"title_canon_sha256":"532dab40a57b46a7f4421c688e9ac4f49618993cd081fcad35302d7265dc8291","abstract_canon_sha256":"5048fd72943530862c9434171f7c874b12d86caadac41d04ae9409dd312b4f74"},"schema_version":"1.0"},"canonical_sha256":"3a866eb4b63360fe8b16def3c5ec0601b17a0dc93efd43ae71bcc007b4b2569f","source":{"kind":"arxiv","id":"1812.10764","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.10764","created_at":"2026-05-17T23:51:57Z"},{"alias_kind":"arxiv_version","alias_value":"1812.10764v2","created_at":"2026-05-17T23:51:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.10764","created_at":"2026-05-17T23:51:57Z"},{"alias_kind":"pith_short_12","alias_value":"HKDG5NFWGNQP","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HKDG5NFWGNQP5CYW","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HKDG5NFW","created_at":"2026-05-18T12:32:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:HKDG5NFWGNQP5CYW33Z4L3AGAG","target":"record","payload":{"canonical_record":{"source":{"id":"1812.10764","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-12-27T16:42:52Z","cross_cats_sorted":[],"title_canon_sha256":"532dab40a57b46a7f4421c688e9ac4f49618993cd081fcad35302d7265dc8291","abstract_canon_sha256":"5048fd72943530862c9434171f7c874b12d86caadac41d04ae9409dd312b4f74"},"schema_version":"1.0"},"canonical_sha256":"3a866eb4b63360fe8b16def3c5ec0601b17a0dc93efd43ae71bcc007b4b2569f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:57.081472Z","signature_b64":"PyON07lJWmywk2kO0VmHudj/iWbgKr+F+gA9jFxx8iO83+h/Xe/xQATcq0Cn60xbhZre5XS+GsSz5fUR3L5xCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a866eb4b63360fe8b16def3c5ec0601b17a0dc93efd43ae71bcc007b4b2569f","last_reissued_at":"2026-05-17T23:51:57.080895Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:57.080895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.10764","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:51:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ymij3c1reolY3Sk1RMkVe0SkFW19wL0kk+C3CYpkm9KBk0bXcx/+FQNFENihl90PHFpxE9B8Ffu20DYXpUD5DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T10:35:59.940027Z"},"content_sha256":"240c30625ad6a182a9511647dba065fbe47c32c3fc1b6e3dff3ccea27251087a","schema_version":"1.0","event_id":"sha256:240c30625ad6a182a9511647dba065fbe47c32c3fc1b6e3dff3ccea27251087a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:HKDG5NFWGNQP5CYW33Z4L3AGAG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An asymptotic expansion for a sum of modified Bessel functions with quadratic argument","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R. B. Paris","submitted_at":"2018-12-27T16:42:52Z","abstract_excerpt":"We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \\[S_\\nu(a)=\\sum_{n\\geq 1} (\\frac{1}{2} an^2)^{-\\nu} K_\\nu(an^2)\\qquad (|\\arg\\,a|<\\pi/2)\\] as the parameter $|a|\\to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $a\\to 0$ through positive values. Nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10764","kind":"arxiv","version":2},"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":"2026-05-17T23:51:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mfUWgrDPIzxu9yctviZMekvCCgbzLHnqOPVqN/aP45R7v8tN5KffM1pz5wtO4mYOTt+mlgauHueTBv+yaAdJDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T10:35:59.940392Z"},"content_sha256":"00485ed214a563618fbeb75a7bdd16d636d4b982d6dbd8fcade49dea44564b1a","schema_version":"1.0","event_id":"sha256:00485ed214a563618fbeb75a7bdd16d636d4b982d6dbd8fcade49dea44564b1a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG/bundle.json","state_url":"https://pith.science/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG/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":"2026-06-12T10:35:59Z","links":{"resolver":"https://pith.science/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG","bundle":"https://pith.science/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG/bundle.json","state":"https://pith.science/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HKDG5NFWGNQP5CYW33Z4L3AGAG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HKDG5NFWGNQP5CYW33Z4L3AGAG","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":"5048fd72943530862c9434171f7c874b12d86caadac41d04ae9409dd312b4f74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-12-27T16:42:52Z","title_canon_sha256":"532dab40a57b46a7f4421c688e9ac4f49618993cd081fcad35302d7265dc8291"},"schema_version":"1.0","source":{"id":"1812.10764","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.10764","created_at":"2026-05-17T23:51:57Z"},{"alias_kind":"arxiv_version","alias_value":"1812.10764v2","created_at":"2026-05-17T23:51:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.10764","created_at":"2026-05-17T23:51:57Z"},{"alias_kind":"pith_short_12","alias_value":"HKDG5NFWGNQP","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HKDG5NFWGNQP5CYW","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HKDG5NFW","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:00485ed214a563618fbeb75a7bdd16d636d4b982d6dbd8fcade49dea44564b1a","target":"graph","created_at":"2026-05-17T23:51:57Z","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":"We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \\[S_\\nu(a)=\\sum_{n\\geq 1} (\\frac{1}{2} an^2)^{-\\nu} K_\\nu(an^2)\\qquad (|\\arg\\,a|<\\pi/2)\\] as the parameter $|a|\\to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $a\\to 0$ through positive values. Nu","authors_text":"R. B. Paris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-12-27T16:42:52Z","title":"An asymptotic expansion for a sum of modified Bessel functions with quadratic argument"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10764","kind":"arxiv","version":2},"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":"sha256:240c30625ad6a182a9511647dba065fbe47c32c3fc1b6e3dff3ccea27251087a","target":"record","created_at":"2026-05-17T23:51:57Z","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":"5048fd72943530862c9434171f7c874b12d86caadac41d04ae9409dd312b4f74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-12-27T16:42:52Z","title_canon_sha256":"532dab40a57b46a7f4421c688e9ac4f49618993cd081fcad35302d7265dc8291"},"schema_version":"1.0","source":{"id":"1812.10764","kind":"arxiv","version":2}},"canonical_sha256":"3a866eb4b63360fe8b16def3c5ec0601b17a0dc93efd43ae71bcc007b4b2569f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a866eb4b63360fe8b16def3c5ec0601b17a0dc93efd43ae71bcc007b4b2569f","first_computed_at":"2026-05-17T23:51:57.080895Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:57.080895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PyON07lJWmywk2kO0VmHudj/iWbgKr+F+gA9jFxx8iO83+h/Xe/xQATcq0Cn60xbhZre5XS+GsSz5fUR3L5xCw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:57.081472Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.10764","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:240c30625ad6a182a9511647dba065fbe47c32c3fc1b6e3dff3ccea27251087a","sha256:00485ed214a563618fbeb75a7bdd16d636d4b982d6dbd8fcade49dea44564b1a"],"state_sha256":"42b8ccf8d34c9640cbb4aac8fdc1d33eb51e87e65c674bddffd461da5ef849f4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sMkQzpD7qg3GszTRZFfbBhARErk7P47esuCWtoZXnO8O0LNvnqTTc6Y9dfWrG3qswtkBdUHPKO/EYqkKXXv9Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T10:35:59.943159Z","bundle_sha256":"2092fb2e75be8ff3c4c74e7bdf7d9d266490433959af3ce87c345d25fb8a56bc"}}