{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:YRYZAMLMJE4PPQ5YTADIZZVZCF","short_pith_number":"pith:YRYZAMLM","schema_version":"1.0","canonical_sha256":"c47190316c4938f7c3b898068ce6b91176037df8f3174cc1bf29a98fd09d20e5","source":{"kind":"arxiv","id":"1110.5308","version":5},"attestation_state":"computed","paper":{"title":"Congruences concerning Jacobi polynomials and Ap\\'ery-like formulae","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Khodabakhsh Hessami Pilehrood, Roberto Tauraso, Tatiana Hessami Pilehrood","submitted_at":"2011-10-24T19:06:49Z","abstract_excerpt":"Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\\sum_{k=0}^{(p-3)/2}\\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\\sum_{k=1}^{(p-1)/2}\\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$."},"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":"1110.5308","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-10-24T19:06:49Z","cross_cats_sorted":[],"title_canon_sha256":"13dbc8ee61dc36306de767eb821c7704e5aba79c044e9f847b7769f44eeaa421","abstract_canon_sha256":"c2c03ba7a7b59749b6216c878653e65b705c23a27daf0a22817aebfb53a6c8d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:52.766022Z","signature_b64":"JmrRjLV83gX0Yq88CqA2ZqVl3RePTWWDLT4uNKOBG1JGUO8sGq/yYsbsJ9Xddzv6z+GR3ihJOn4tV0BMu8VUBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c47190316c4938f7c3b898068ce6b91176037df8f3174cc1bf29a98fd09d20e5","last_reissued_at":"2026-05-18T03:43:52.765583Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:52.765583Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Congruences concerning Jacobi polynomials and Ap\\'ery-like formulae","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Khodabakhsh Hessami Pilehrood, Roberto Tauraso, Tatiana Hessami Pilehrood","submitted_at":"2011-10-24T19:06:49Z","abstract_excerpt":"Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\\sum_{k=0}^{(p-3)/2}\\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\\sum_{k=1}^{(p-1)/2}\\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5308","kind":"arxiv","version":5},"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":"1110.5308","created_at":"2026-05-18T03:43:52.765648+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.5308v5","created_at":"2026-05-18T03:43:52.765648+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.5308","created_at":"2026-05-18T03:43:52.765648+00:00"},{"alias_kind":"pith_short_12","alias_value":"YRYZAMLMJE4P","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"YRYZAMLMJE4PPQ5Y","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"YRYZAMLM","created_at":"2026-05-18T12:26:47.523578+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/YRYZAMLMJE4PPQ5YTADIZZVZCF","json":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF.json","graph_json":"https://pith.science/api/pith-number/YRYZAMLMJE4PPQ5YTADIZZVZCF/graph.json","events_json":"https://pith.science/api/pith-number/YRYZAMLMJE4PPQ5YTADIZZVZCF/events.json","paper":"https://pith.science/paper/YRYZAMLM"},"agent_actions":{"view_html":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF","download_json":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF.json","view_paper":"https://pith.science/paper/YRYZAMLM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.5308&json=true","fetch_graph":"https://pith.science/api/pith-number/YRYZAMLMJE4PPQ5YTADIZZVZCF/graph.json","fetch_events":"https://pith.science/api/pith-number/YRYZAMLMJE4PPQ5YTADIZZVZCF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF/action/storage_attestation","attest_author":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF/action/author_attestation","sign_citation":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF/action/citation_signature","submit_replication":"https://pith.science/pith/YRYZAMLMJE4PPQ5YTADIZZVZCF/action/replication_record"}},"created_at":"2026-05-18T03:43:52.765648+00:00","updated_at":"2026-05-18T03:43:52.765648+00:00"}