{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2QZBABESHJ5F5EM4VW653JEI3C","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":"3b8dfb2aaa1500de7eeb94d5dcf0ca41149a06686b66e02aff7193941af50ce3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-16T00:48:00Z","title_canon_sha256":"d2a496bae386987709f203f4e759495fbfff4f0caf24e2cbf028a3ad5e39a485"},"schema_version":"1.0","source":{"id":"1507.04425","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04425","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04425v1","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04425","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"pith_short_12","alias_value":"2QZBABESHJ5F","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2QZBABESHJ5F5EM4","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2QZBABES","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:fa4853bcbf4e882019656de32e909ed5da8ed60277ca39eaa090b16399d3251d","target":"graph","created_at":"2026-05-18T01:36:47Z","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 define the normalized Eisenstein series $\\mathcal{P}$, $e$, and $\\mathcal{Q}$ associated with $\\Gamma_0(2),$ and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on $\\Gamma_0(2)$ and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a for","authors_text":"Heekyoung Hahn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-16T00:48:00Z","title":"Eisenstein series associated with $\\Gamma_0(2)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04425","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":"sha256:172896f7df6b146269fcef36461c9568893b4f4e49affe768710dd6bc337bea8","target":"record","created_at":"2026-05-18T01:36:47Z","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":"3b8dfb2aaa1500de7eeb94d5dcf0ca41149a06686b66e02aff7193941af50ce3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-07-16T00:48:00Z","title_canon_sha256":"d2a496bae386987709f203f4e759495fbfff4f0caf24e2cbf028a3ad5e39a485"},"schema_version":"1.0","source":{"id":"1507.04425","kind":"arxiv","version":1}},"canonical_sha256":"d4321004923a7a5e919cadbddda488d8978897342a1958351d5aead6f0364e3f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d4321004923a7a5e919cadbddda488d8978897342a1958351d5aead6f0364e3f","first_computed_at":"2026-05-18T01:36:47.427308Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:47.427308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WlClTRT84wH9IDAIphyn2IF3lX5lTo5tpQ2nkX1PtqxOuASdeWxnlVuhRFSB0OlLsP/nyKm/RTT2xrWdsX8sAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:47.427929Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.04425","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:172896f7df6b146269fcef36461c9568893b4f4e49affe768710dd6bc337bea8","sha256:fa4853bcbf4e882019656de32e909ed5da8ed60277ca39eaa090b16399d3251d"],"state_sha256":"d0a3fdfb50c2579fa0c45cefc37148dc5c10a5a68afca79c3070e2323063c965"}