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Among others, we prove that $$ \\sum_{k=0}^{[n/4]}(-1)^k {n\\choose 4k}\\frac{B_{n-4k}(z) }{2^{6k}} =\\frac{1}{2^{n+1}}\\sum_{k=0}^{n} (-1)^k \\frac{1+i^k}{(1+i)^k} {n\\choose k}{B_{n-k}(2z)} $$ and $$ \\sum_{k=1}^{n} 2^{2k-1} {2n\\choose 2k-1} B_{2k-1}(z) =\n\\sum_{k=1}^n k \\, 2^{2k} {2n\\choose 2k} E_{2k-1}(z). $$ Applications of our results lead to formulas for Bernoulli and Euler numbers, like, for instance, $$ n E_{n-1} =\\sum_{k=1}^{[n/2]} \\frac{2^{2k}-1}{k} (2^{2k}-2^n){n\\choose 2k-1} B_{2k}B_{n-2k}. $$","authors_text":"Horst Alzer, Semyon Yakubovich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-19T13:18:52Z","title":"Identities involving Bernoulli and Euler polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07127","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":"sha256:5c3f2a7a81ac436d7b6323d61dacf45ce601f86c2a5ef72ca1adbb3d7bc619f6","target":"record","created_at":"2026-05-18T00:32:26Z","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":"78cc170a7fde2a6066e5846317f1739176ce6ba1a44af11d0217b86f764efd05","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-10-19T13:18:52Z","title_canon_sha256":"735d3bb4f9aa7feaec2bbd6a96bf9193b8d7992990f8644bc273ed9d809c2f58"},"schema_version":"1.0","source":{"id":"1710.07127","kind":"arxiv","version":1}},"canonical_sha256":"1dc95ed2d817060d396c19038ef354575857b193c0ad13ca552f310bfb770b29","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1dc95ed2d817060d396c19038ef354575857b193c0ad13ca552f310bfb770b29","first_computed_at":"2026-05-18T00:32:26.927449Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:26.927449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bckx0TUwCSyyDRfFJiiOFdmBHmneQFIdExjuSusyORJHVmx69vwtGpOyHB2x7oFKstWCLifD7L8o4LBRKdQmDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:26.928102Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.07127","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c3f2a7a81ac436d7b6323d61dacf45ce601f86c2a5ef72ca1adbb3d7bc619f6","sha256:af50923493489e1879d4bd86a437da730f9de074cc2d3878288fa032dfa30fac"],"state_sha256":"9917fc8a9e8d862bad24005b4d2b2578d2cef16a9510757493fd2e92b8155e63"}