{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:OOW2VQRAYZ3WYGRH3DRBWMHFEX","short_pith_number":"pith:OOW2VQRA","canonical_record":{"source":{"id":"2607.10135","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-11T05:49:28Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"6bae0eccd3a7b618e538b46b1b6a3b9baf5e447a51854a1d30093e0e617890c8","abstract_canon_sha256":"d9ad644972da51d6cee03961e3b77b40195a71b8e6b6189589d93dac560da292"},"schema_version":"1.0"},"canonical_sha256":"73adaac220c6776c1a27d8e21b30e525c306fb7cf495356992ea5aa4e4d16c9e","source":{"kind":"arxiv","id":"2607.10135","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10135","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10135v1","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10135","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"OOW2VQRAYZ3W","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_16","alias_value":"OOW2VQRAYZ3WYGRH","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_8","alias_value":"OOW2VQRA","created_at":"2026-07-14T01:20:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:OOW2VQRAYZ3WYGRH3DRBWMHFEX","target":"record","payload":{"canonical_record":{"source":{"id":"2607.10135","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-11T05:49:28Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"6bae0eccd3a7b618e538b46b1b6a3b9baf5e447a51854a1d30093e0e617890c8","abstract_canon_sha256":"d9ad644972da51d6cee03961e3b77b40195a71b8e6b6189589d93dac560da292"},"schema_version":"1.0"},"canonical_sha256":"73adaac220c6776c1a27d8e21b30e525c306fb7cf495356992ea5aa4e4d16c9e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:20:27.070540Z","signature_b64":"thgbHw6sIwmaGhawMDkkAGC5WMLdC09gQbaPUU7WeYSfTMLNNJIyeq4VTSjUNrZc2BSDljWULG/kCQU6ior6CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73adaac220c6776c1a27d8e21b30e525c306fb7cf495356992ea5aa4e4d16c9e","last_reissued_at":"2026-07-14T01:20:27.069719Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:20:27.069719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.10135","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":"2026-07-14T01:20:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uvxr4uxYjinmSY8IgnEJ6kVy5WbSq556skTgOcRAMy8xn2WtEwKfCmzcbHr/NWQUGZDozsy3vqMDfMTFgFv3DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:52:08.336166Z"},"content_sha256":"97b13f9c6561fa53d30bb1660882cb2ac7f7dcafb77fe91f0cb0210a684a1cbd","schema_version":"1.0","event_id":"sha256:97b13f9c6561fa53d30bb1660882cb2ac7f7dcafb77fe91f0cb0210a684a1cbd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:OOW2VQRAYZ3WYGRH3DRBWMHFEX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a Class of Hypergeometric Sums via Recurrences and Product Binomial-Harmonic Identities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Narendra Bhandari","submitted_at":"2026-07-11T05:49:28Z","abstract_excerpt":"We study hypergeometric series with coefficients \\(\n  a_n(\\alpha)=\\frac{(\\alpha)_n(1-\\alpha)_n}{(n!)^2},\\) where \\(0< \\alpha < 1\\). The main idea is to introduce a shift parameter in the linear denominator and consider \\[\n  \\Phi_{m,\\varepsilon}(\\lambda;\\alpha)\n  =\n  \\sum_{n=0}^{\\infty}\\varepsilon^n\n  \\frac{a_n(\\alpha)}{n+m+1+\\lambda},\n  \\qquad \\varepsilon\\in\\{1,-1\\}. \\] Expanding this expression in powers of \\(\\lambda\\) produces sums with denominator powers \\((n+m+1)^{-K}\\). We first discuss analytic interpolation in the denominator exponent and explain why positive integer exponents lead to t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10135","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/2607.10135/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":"2026-07-14T01:20:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GmWViwEBWyahZ8FVfajhx8CSwgY9t8g2+UzWz6B2BFoN9FbIyPeks2hszJhr86TPMolItFrFrf/Jhj/1diU3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:52:08.336710Z"},"content_sha256":"564d5c106ac81335cd6ff4fe9645fc146298c3d804a7612dd595bcdc2a5af489","schema_version":"1.0","event_id":"sha256:564d5c106ac81335cd6ff4fe9645fc146298c3d804a7612dd595bcdc2a5af489"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX/bundle.json","state_url":"https://pith.science/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX/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":"2026-08-03T16:52:08Z","links":{"resolver":"https://pith.science/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX","bundle":"https://pith.science/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX/bundle.json","state":"https://pith.science/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OOW2VQRAYZ3WYGRH3DRBWMHFEX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:OOW2VQRAYZ3WYGRH3DRBWMHFEX","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":"d9ad644972da51d6cee03961e3b77b40195a71b8e6b6189589d93dac560da292","cross_cats_sorted":["math.CA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-11T05:49:28Z","title_canon_sha256":"6bae0eccd3a7b618e538b46b1b6a3b9baf5e447a51854a1d30093e0e617890c8"},"schema_version":"1.0","source":{"id":"2607.10135","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10135","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10135v1","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10135","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_12","alias_value":"OOW2VQRAYZ3W","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_16","alias_value":"OOW2VQRAYZ3WYGRH","created_at":"2026-07-14T01:20:27Z"},{"alias_kind":"pith_short_8","alias_value":"OOW2VQRA","created_at":"2026-07-14T01:20:27Z"}],"graph_snapshots":[{"event_id":"sha256:564d5c106ac81335cd6ff4fe9645fc146298c3d804a7612dd595bcdc2a5af489","target":"graph","created_at":"2026-07-14T01:20:27Z","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/2607.10135/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study hypergeometric series with coefficients \\(\n  a_n(\\alpha)=\\frac{(\\alpha)_n(1-\\alpha)_n}{(n!)^2},\\) where \\(0< \\alpha < 1\\). The main idea is to introduce a shift parameter in the linear denominator and consider \\[\n  \\Phi_{m,\\varepsilon}(\\lambda;\\alpha)\n  =\n  \\sum_{n=0}^{\\infty}\\varepsilon^n\n  \\frac{a_n(\\alpha)}{n+m+1+\\lambda},\n  \\qquad \\varepsilon\\in\\{1,-1\\}. \\] Expanding this expression in powers of \\(\\lambda\\) produces sums with denominator powers \\((n+m+1)^{-K}\\). We first discuss analytic interpolation in the denominator exponent and explain why positive integer exponents lead to t","authors_text":"Narendra Bhandari","cross_cats":["math.CA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-11T05:49:28Z","title":"On a Class of Hypergeometric Sums via Recurrences and Product Binomial-Harmonic Identities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10135","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":"sha256:97b13f9c6561fa53d30bb1660882cb2ac7f7dcafb77fe91f0cb0210a684a1cbd","target":"record","created_at":"2026-07-14T01:20:27Z","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":"d9ad644972da51d6cee03961e3b77b40195a71b8e6b6189589d93dac560da292","cross_cats_sorted":["math.CA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-11T05:49:28Z","title_canon_sha256":"6bae0eccd3a7b618e538b46b1b6a3b9baf5e447a51854a1d30093e0e617890c8"},"schema_version":"1.0","source":{"id":"2607.10135","kind":"arxiv","version":1}},"canonical_sha256":"73adaac220c6776c1a27d8e21b30e525c306fb7cf495356992ea5aa4e4d16c9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73adaac220c6776c1a27d8e21b30e525c306fb7cf495356992ea5aa4e4d16c9e","first_computed_at":"2026-07-14T01:20:27.069719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-14T01:20:27.069719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"thgbHw6sIwmaGhawMDkkAGC5WMLdC09gQbaPUU7WeYSfTMLNNJIyeq4VTSjUNrZc2BSDljWULG/kCQU6ior6CQ==","signature_status":"signed_v1","signed_at":"2026-07-14T01:20:27.070540Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.10135","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97b13f9c6561fa53d30bb1660882cb2ac7f7dcafb77fe91f0cb0210a684a1cbd","sha256:564d5c106ac81335cd6ff4fe9645fc146298c3d804a7612dd595bcdc2a5af489"],"state_sha256":"d0a2df7ca960fdb443db65ea8dd3f8a4bc9e41e203db91251d6b5b366d7ee7af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bR1udDR6qM8807v86LyPrqRmj7MkP8gA3uWWZ+PLzUoz5BVVAVXdbFW12TVorx17nMME1E1tHLsTu7475tvUAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T16:52:08.343382Z","bundle_sha256":"df6a45bdfd112726866d2efc2a4f37e19e1b581e4ce4164f65de85f1870f3c9a"}}