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We determine the complete asymptotic expansion, in powers of $N^{-1}$, of the binomial mass $\\Pr\\{X=\\nu+r\\}$ -- a quotient of gamma functions -- uniformly for $p$ in compact subintervals of $(0,1)$ and for bounded $r$. Because the mean $Np$ is not a lattice point, the coefficients cannot be constants: they are Bernoulli polynomials evaluated at the oscillating fractional displacement $h_N=\\nu-Np\\in[0,1)$, and are given in closed form to all ord"},"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":"2607.19844","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-22T07:30:15Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"feeee16a5e428b8fc6e689f9641e7a543a5783d58c3e9d858a3d89d1f1731369","abstract_canon_sha256":"4125495670b01e401c3ba46f7f52b39d51588edfe39b5c96f2250c44a4a98129"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:24:03.270336Z","signature_b64":"zqI5njcB2vkSHZgn1EuxLFpqsZ3lu39HvR0WJaby9nd5qI5mjZDKYPXQ5acrq5RndyAJ2yMBiQ9Q6uSQERjbCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63cbb43fc5d4f2831a732b05099b65fa794815b59b55f02d58d3b55436490dae","last_reissued_at":"2026-07-23T01:24:03.269443Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:24:03.269443Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Neven Elezovi\\'c","submitted_at":"2026-07-22T07:30:15Z","abstract_excerpt":"Let $X\\sim Bin(N,p)$ with $0<p<1$ and $q=1-p$, let $\\nu=\\lceil Np\\rceil$ be the first lattice point not below the mean, and let $r$ be a fixed integer. We determine the complete asymptotic expansion, in powers of $N^{-1}$, of the binomial mass $\\Pr\\{X=\\nu+r\\}$ -- a quotient of gamma functions -- uniformly for $p$ in compact subintervals of $(0,1)$ and for bounded $r$. Because the mean $Np$ is not a lattice point, the coefficients cannot be constants: they are Bernoulli polynomials evaluated at the oscillating fractional displacement $h_N=\\nu-Np\\in[0,1)$, and are given in closed form to all ord"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19844","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/2607.19844/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.19844","created_at":"2026-07-23T01:24:03.269913+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.19844v1","created_at":"2026-07-23T01:24:03.269913+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19844","created_at":"2026-07-23T01:24:03.269913+00:00"},{"alias_kind":"pith_short_12","alias_value":"MPF3IP6F2TZI","created_at":"2026-07-23T01:24:03.269913+00:00"},{"alias_kind":"pith_short_16","alias_value":"MPF3IP6F2TZIGGTT","created_at":"2026-07-23T01:24:03.269913+00:00"},{"alias_kind":"pith_short_8","alias_value":"MPF3IP6F","created_at":"2026-07-23T01:24:03.269913+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.06232","citing_title":"The mean absolute deviation of the classical discrete distributions: collapse identities, complete asymptotic expansions, and enveloping series","ref_index":10,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J","json":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J.json","graph_json":"https://pith.science/api/pith-number/MPF3IP6F2TZIGGTTFMCQTG3F7J/graph.json","events_json":"https://pith.science/api/pith-number/MPF3IP6F2TZIGGTTFMCQTG3F7J/events.json","paper":"https://pith.science/paper/MPF3IP6F"},"agent_actions":{"view_html":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J","download_json":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J.json","view_paper":"https://pith.science/paper/MPF3IP6F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.19844&json=true","fetch_graph":"https://pith.science/api/pith-number/MPF3IP6F2TZIGGTTFMCQTG3F7J/graph.json","fetch_events":"https://pith.science/api/pith-number/MPF3IP6F2TZIGGTTFMCQTG3F7J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/action/storage_attestation","attest_author":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/action/author_attestation","sign_citation":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/action/citation_signature","submit_replication":"https://pith.science/pith/MPF3IP6F2TZIGGTTFMCQTG3F7J/action/replication_record"}},"created_at":"2026-07-23T01:24:03.269913+00:00","updated_at":"2026-07-23T01:24:03.269913+00:00"}