{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:WWNGYSER2MOF4PMHFCLMHHDNQ3","short_pith_number":"pith:WWNGYSER","canonical_record":{"source":{"id":"2412.02462","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-03T14:04:50Z","cross_cats_sorted":["math.CO","math.CV"],"title_canon_sha256":"bd066e07704e805fb234a83857b7fa1375efa77fbd51e83169b538bfcb3378b9","abstract_canon_sha256":"4a83cfec0b154b3b1e801a7959fdb22c311fa33b5309d125f817279327a389c1"},"schema_version":"1.0"},"canonical_sha256":"b59a6c4891d31c5e3d872896c39c6d86efe6d9169dbc8346405e2a9e5957988c","source":{"kind":"arxiv","id":"2412.02462","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.02462","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"arxiv_version","alias_value":"2412.02462v1","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.02462","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"pith_short_12","alias_value":"WWNGYSER2MOF","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"pith_short_16","alias_value":"WWNGYSER2MOF4PMH","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"pith_short_8","alias_value":"WWNGYSER","created_at":"2026-07-05T09:43:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:WWNGYSER2MOF4PMHFCLMHHDNQ3","target":"record","payload":{"canonical_record":{"source":{"id":"2412.02462","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-03T14:04:50Z","cross_cats_sorted":["math.CO","math.CV"],"title_canon_sha256":"bd066e07704e805fb234a83857b7fa1375efa77fbd51e83169b538bfcb3378b9","abstract_canon_sha256":"4a83cfec0b154b3b1e801a7959fdb22c311fa33b5309d125f817279327a389c1"},"schema_version":"1.0"},"canonical_sha256":"b59a6c4891d31c5e3d872896c39c6d86efe6d9169dbc8346405e2a9e5957988c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:43:50.248455Z","signature_b64":"g4aguHmU7VevSvO05iNbgzBNzJGWZbdKLKpe++rsCJ9Xtrpjsvzgkp8SjEfW64tGG9VRnHLyqMmpWzW0mWULAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b59a6c4891d31c5e3d872896c39c6d86efe6d9169dbc8346405e2a9e5957988c","last_reissued_at":"2026-07-05T09:43:50.247933Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:43:50.247933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.02462","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":"2026-07-05T09:43:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EAy9puln4ZRwP/GQHsCQ1ZQ8cQtbQK5YCgYCaLyZEW7BjXm7QDeDRlht4+zgQeC/OjagOoOX9QhmTvGBPC/qBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T14:32:56.563179Z"},"content_sha256":"18a280840cf56558ce2396489f45876c1867a7c27127a499aaffefb78d37f86d","schema_version":"1.0","event_id":"sha256:18a280840cf56558ce2396489f45876c1867a7c27127a499aaffefb78d37f86d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:WWNGYSER2MOF4PMHFCLMHHDNQ3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On series expansions of zeros of the deformed exponential function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.CV"],"primary_cat":"math.CA","authors_text":"Alexey Kuznetsov","submitted_at":"2024-12-03T14:04:50Z","abstract_excerpt":"For $q \\in (0, 1)$, the deformed exponential function $f(x) = \\sum_{n \\geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\\{x_k(q)\\}_{k \\geq 1}$. In this paper, we analyze the series expansions of $-x_k(q)/k$ and $k/x_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)/Q_n(k)$ and $\\widehat{P}_n(k)/Q_n(k)$, where $Q_n(k) \\in {\\mathbb Z}[k]$ is explicitly defined and the polynomials $P_n(k), \\widehat{P}_n(k)\\in {\\mathbb Z}[k]$ can be computed recursively. We provide explicit formulas for the l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.02462","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.02462/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:43:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KolNjtvryL1lgk/O3e1HyV12eMwUt/E1xJMgejx0/uYOO7kX+W3iQGRDWiwIouLhbY6DHM0VjUcvq3ucFMWHAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T14:32:56.563522Z"},"content_sha256":"36a66393922d67fb21ede31b7d719943bd583f5f3f348c35bf55a10408364ccd","schema_version":"1.0","event_id":"sha256:36a66393922d67fb21ede31b7d719943bd583f5f3f348c35bf55a10408364ccd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3/bundle.json","state_url":"https://pith.science/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3/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":"2026-08-20T14:32:56Z","links":{"resolver":"https://pith.science/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3","bundle":"https://pith.science/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3/bundle.json","state":"https://pith.science/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WWNGYSER2MOF4PMHFCLMHHDNQ3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:WWNGYSER2MOF4PMHFCLMHHDNQ3","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":"4a83cfec0b154b3b1e801a7959fdb22c311fa33b5309d125f817279327a389c1","cross_cats_sorted":["math.CO","math.CV"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-03T14:04:50Z","title_canon_sha256":"bd066e07704e805fb234a83857b7fa1375efa77fbd51e83169b538bfcb3378b9"},"schema_version":"1.0","source":{"id":"2412.02462","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.02462","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"arxiv_version","alias_value":"2412.02462v1","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.02462","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"pith_short_12","alias_value":"WWNGYSER2MOF","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"pith_short_16","alias_value":"WWNGYSER2MOF4PMH","created_at":"2026-07-05T09:43:50Z"},{"alias_kind":"pith_short_8","alias_value":"WWNGYSER","created_at":"2026-07-05T09:43:50Z"}],"graph_snapshots":[{"event_id":"sha256:36a66393922d67fb21ede31b7d719943bd583f5f3f348c35bf55a10408364ccd","target":"graph","created_at":"2026-07-05T09:43:50Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2412.02462/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For $q \\in (0, 1)$, the deformed exponential function $f(x) = \\sum_{n \\geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\\{x_k(q)\\}_{k \\geq 1}$. In this paper, we analyze the series expansions of $-x_k(q)/k$ and $k/x_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)/Q_n(k)$ and $\\widehat{P}_n(k)/Q_n(k)$, where $Q_n(k) \\in {\\mathbb Z}[k]$ is explicitly defined and the polynomials $P_n(k), \\widehat{P}_n(k)\\in {\\mathbb Z}[k]$ can be computed recursively. We provide explicit formulas for the l","authors_text":"Alexey Kuznetsov","cross_cats":["math.CO","math.CV"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-03T14:04:50Z","title":"On series expansions of zeros of the deformed exponential function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.02462","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":"sha256:18a280840cf56558ce2396489f45876c1867a7c27127a499aaffefb78d37f86d","target":"record","created_at":"2026-07-05T09:43:50Z","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":"4a83cfec0b154b3b1e801a7959fdb22c311fa33b5309d125f817279327a389c1","cross_cats_sorted":["math.CO","math.CV"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-03T14:04:50Z","title_canon_sha256":"bd066e07704e805fb234a83857b7fa1375efa77fbd51e83169b538bfcb3378b9"},"schema_version":"1.0","source":{"id":"2412.02462","kind":"arxiv","version":1}},"canonical_sha256":"b59a6c4891d31c5e3d872896c39c6d86efe6d9169dbc8346405e2a9e5957988c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b59a6c4891d31c5e3d872896c39c6d86efe6d9169dbc8346405e2a9e5957988c","first_computed_at":"2026-07-05T09:43:50.247933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:43:50.247933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g4aguHmU7VevSvO05iNbgzBNzJGWZbdKLKpe++rsCJ9Xtrpjsvzgkp8SjEfW64tGG9VRnHLyqMmpWzW0mWULAg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:43:50.248455Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.02462","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18a280840cf56558ce2396489f45876c1867a7c27127a499aaffefb78d37f86d","sha256:36a66393922d67fb21ede31b7d719943bd583f5f3f348c35bf55a10408364ccd"],"state_sha256":"8a5c13be31a3b485392ccefaa7555f5146d671e1cb7a93d52af3d8ef51201c5c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yQiLETUWobU867djvdOE+5Avo+AAq/csSOAJSUI2T0ZE5Yfve0fM+0vynov4PNfCvBeRKr7l8J08/8EvYoxFAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T14:32:56.567002Z","bundle_sha256":"d349080f1513ae94320dd87b8d316960e6b1287fb68c28809fb3e75252162c7e"}}