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We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\\hat L_n^\\alpha$, the polynomials $\\mathcal L_n^\\alpha=n!L_n^\\alpha/(1+\\alpha)_n$ (so that $\\mathcal L_n^\\alpha(0)=1$), the standard Laguerre polynomials $(L_n^\\alpha)_n$ and the Brenke normalization $L_n^\\alpha/(1+\\a"},"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":"2507.22425","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2025-07-30T07:06:41Z","cross_cats_sorted":[],"title_canon_sha256":"0c38a838f53fb1d18148416751b43708aa187b3056c69466a5845357e2d93051","abstract_canon_sha256":"1e29d9ba32f1c053a7499c050d466d3d7b5e4656287af688e0f83b739b59dbba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:45:41.779977Z","signature_b64":"H73HCK6MzkaA2n8DtCZZEH90Zihd4HzOLOjIuhgy79FUQSylkW9QQuyEm6r1G1XuKGwKsnaljRA9Ej4iQTkXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02cc6425b89cf92f45a191aee7e70c42e675281bfd63fe085b2e1119f8fe12bb","last_reissued_at":"2026-07-05T11:45:41.779477Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:45:41.779477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zeros of linear combinations of Laguerre polynomials","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Antonio J. Dur\\'an","submitted_at":"2025-07-30T07:06:41Z","abstract_excerpt":"We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\\sum_{j=0}^K\\gamma_j\\tilde L^\\alpha_{n-j}(x),\\quad n\\ge K, $$ where $\\gamma_j$, $j=0,\\cdots ,K$, are real numbers with $\\gamma_0=1$, $\\gamma_K\\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\\hat L_n^\\alpha$, the polynomials $\\mathcal L_n^\\alpha=n!L_n^\\alpha/(1+\\alpha)_n$ (so that $\\mathcal L_n^\\alpha(0)=1$), the standard Laguerre polynomials $(L_n^\\alpha)_n$ and the Brenke normalization $L_n^\\alpha/(1+\\a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.22425","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/2507.22425/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":"2507.22425","created_at":"2026-07-05T11:45:41.779535+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.22425v1","created_at":"2026-07-05T11:45:41.779535+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.22425","created_at":"2026-07-05T11:45:41.779535+00:00"},{"alias_kind":"pith_short_12","alias_value":"ALGGIJNYTT4S","created_at":"2026-07-05T11:45:41.779535+00:00"},{"alias_kind":"pith_short_16","alias_value":"ALGGIJNYTT4S6RNB","created_at":"2026-07-05T11:45:41.779535+00:00"},{"alias_kind":"pith_short_8","alias_value":"ALGGIJNY","created_at":"2026-07-05T11:45:41.779535+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/ALGGIJNYTT4S6RNBSGXOPZYMIL","json":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL.json","graph_json":"https://pith.science/api/pith-number/ALGGIJNYTT4S6RNBSGXOPZYMIL/graph.json","events_json":"https://pith.science/api/pith-number/ALGGIJNYTT4S6RNBSGXOPZYMIL/events.json","paper":"https://pith.science/paper/ALGGIJNY"},"agent_actions":{"view_html":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL","download_json":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL.json","view_paper":"https://pith.science/paper/ALGGIJNY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.22425&json=true","fetch_graph":"https://pith.science/api/pith-number/ALGGIJNYTT4S6RNBSGXOPZYMIL/graph.json","fetch_events":"https://pith.science/api/pith-number/ALGGIJNYTT4S6RNBSGXOPZYMIL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL/action/storage_attestation","attest_author":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL/action/author_attestation","sign_citation":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL/action/citation_signature","submit_replication":"https://pith.science/pith/ALGGIJNYTT4S6RNBSGXOPZYMIL/action/replication_record"}},"created_at":"2026-07-05T11:45:41.779535+00:00","updated_at":"2026-07-05T11:45:41.779535+00:00"}