{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:5HEAL4SJTRPVJICZCRJMICXVDT","short_pith_number":"pith:5HEAL4SJ","canonical_record":{"source":{"id":"1111.4880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-11-21T14:25:09Z","cross_cats_sorted":[],"title_canon_sha256":"c871c502492af127c7e04769ed977ca87d88df681303779ded579345b66183c6","abstract_canon_sha256":"7af3340a02652ef627dd77622d4cf2aca1445aa0f9e535061dbe9ed5ad380cc6"},"schema_version":"1.0"},"canonical_sha256":"e9c805f2499c5f54a0591452c40af51cf49d65b2f346347eb3db63037bbb5c35","source":{"kind":"arxiv","id":"1111.4880","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.4880","created_at":"2026-05-18T00:00:37Z"},{"alias_kind":"arxiv_version","alias_value":"1111.4880v1","created_at":"2026-05-18T00:00:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.4880","created_at":"2026-05-18T00:00:37Z"},{"alias_kind":"pith_short_12","alias_value":"5HEAL4SJTRPV","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5HEAL4SJTRPVJICZ","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5HEAL4SJ","created_at":"2026-05-18T12:26:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:5HEAL4SJTRPVJICZCRJMICXVDT","target":"record","payload":{"canonical_record":{"source":{"id":"1111.4880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-11-21T14:25:09Z","cross_cats_sorted":[],"title_canon_sha256":"c871c502492af127c7e04769ed977ca87d88df681303779ded579345b66183c6","abstract_canon_sha256":"7af3340a02652ef627dd77622d4cf2aca1445aa0f9e535061dbe9ed5ad380cc6"},"schema_version":"1.0"},"canonical_sha256":"e9c805f2499c5f54a0591452c40af51cf49d65b2f346347eb3db63037bbb5c35","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:37.432171Z","signature_b64":"zL/+2JOryiKHEGzwq/ysirEAA4pWSgeTsQK5dqgujobFKhDEJhUM91m1OBoC2/owTne2ar0aB8OACF1/FsOUAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9c805f2499c5f54a0591452c40af51cf49d65b2f346347eb3db63037bbb5c35","last_reissued_at":"2026-05-18T00:00:37.431605Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:37.431605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.4880","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":"2026-05-18T00:00:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"brCxjmmynh5pxgmCOava7IDPz7Y3PLgWlGPoeCB3BCDTs5A431aSm3ZBiUFheTzIfWF6rbTXNjWKlL8pB1uKAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:45:49.300265Z"},"content_sha256":"824d65db30b977d2c4e28e7761fb8e7afbfa6de2aba57b5042fad984b6da56d1","schema_version":"1.0","event_id":"sha256:824d65db30b977d2c4e28e7761fb8e7afbfa6de2aba57b5042fad984b6da56d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:5HEAL4SJTRPVJICZCRJMICXVDT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Yilmaz Simsek","submitted_at":"2011-11-21T14:25:09Z","abstract_excerpt":"The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4880","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":""},"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":"2026-05-18T00:00:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X7TSYYmXgyyVeWuH89M1KG7eyzbEeHNMI4jAI44EenJSMqu3RyXLNaxsDUciJaxqUMK7lStgTZbkmNGUraX5CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T11:45:49.300634Z"},"content_sha256":"eb66eb5b62a27459c66fcd4385c33ca9f0395d90926cbe53a1baf9c49ea25e39","schema_version":"1.0","event_id":"sha256:eb66eb5b62a27459c66fcd4385c33ca9f0395d90926cbe53a1baf9c49ea25e39"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5HEAL4SJTRPVJICZCRJMICXVDT/bundle.json","state_url":"https://pith.science/pith/5HEAL4SJTRPVJICZCRJMICXVDT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5HEAL4SJTRPVJICZCRJMICXVDT/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":"2026-06-27T11:45:49Z","links":{"resolver":"https://pith.science/pith/5HEAL4SJTRPVJICZCRJMICXVDT","bundle":"https://pith.science/pith/5HEAL4SJTRPVJICZCRJMICXVDT/bundle.json","state":"https://pith.science/pith/5HEAL4SJTRPVJICZCRJMICXVDT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5HEAL4SJTRPVJICZCRJMICXVDT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5HEAL4SJTRPVJICZCRJMICXVDT","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":"7af3340a02652ef627dd77622d4cf2aca1445aa0f9e535061dbe9ed5ad380cc6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-11-21T14:25:09Z","title_canon_sha256":"c871c502492af127c7e04769ed977ca87d88df681303779ded579345b66183c6"},"schema_version":"1.0","source":{"id":"1111.4880","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.4880","created_at":"2026-05-18T00:00:37Z"},{"alias_kind":"arxiv_version","alias_value":"1111.4880v1","created_at":"2026-05-18T00:00:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.4880","created_at":"2026-05-18T00:00:37Z"},{"alias_kind":"pith_short_12","alias_value":"5HEAL4SJTRPV","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_16","alias_value":"5HEAL4SJTRPVJICZ","created_at":"2026-05-18T12:26:20Z"},{"alias_kind":"pith_short_8","alias_value":"5HEAL4SJ","created_at":"2026-05-18T12:26:20Z"}],"graph_snapshots":[{"event_id":"sha256:eb66eb5b62a27459c66fcd4385c33ca9f0395d90926cbe53a1baf9c49ea25e39","target":"graph","created_at":"2026-05-18T00:00:37Z","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":"The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on th","authors_text":"Yilmaz Simsek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-11-21T14:25:09Z","title":"Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4880","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":"sha256:824d65db30b977d2c4e28e7761fb8e7afbfa6de2aba57b5042fad984b6da56d1","target":"record","created_at":"2026-05-18T00:00:37Z","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":"7af3340a02652ef627dd77622d4cf2aca1445aa0f9e535061dbe9ed5ad380cc6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-11-21T14:25:09Z","title_canon_sha256":"c871c502492af127c7e04769ed977ca87d88df681303779ded579345b66183c6"},"schema_version":"1.0","source":{"id":"1111.4880","kind":"arxiv","version":1}},"canonical_sha256":"e9c805f2499c5f54a0591452c40af51cf49d65b2f346347eb3db63037bbb5c35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e9c805f2499c5f54a0591452c40af51cf49d65b2f346347eb3db63037bbb5c35","first_computed_at":"2026-05-18T00:00:37.431605Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:37.431605Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zL/+2JOryiKHEGzwq/ysirEAA4pWSgeTsQK5dqgujobFKhDEJhUM91m1OBoC2/owTne2ar0aB8OACF1/FsOUAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:37.432171Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.4880","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:824d65db30b977d2c4e28e7761fb8e7afbfa6de2aba57b5042fad984b6da56d1","sha256:eb66eb5b62a27459c66fcd4385c33ca9f0395d90926cbe53a1baf9c49ea25e39"],"state_sha256":"a4c8f1933d868df5f587c290a96bae55185f8155c2a1417cc58d4a2ebb6831ad"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sG32cxmbr5vUQOOeDexDz0mEoBEp2n++0CP7A7qebU4p7Gupb6hm89OYoL5jNKq87kLabDnAbs0NH5jCCxdmDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T11:45:49.302705Z","bundle_sha256":"5d7dc98a8986465d793f5d27d69f9059c7ab40df5a2c120c71450649e9eadc9c"}}