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We show that the operator $Q = p L_\\mu^2 - q L_\\mu$ has polynomial eigenfunctions if and only if $\\mu$ is a free Meixner distribution. The only time $Q$ has orthogonal polynomial eigenfunctions is if $\\mu$ is a semicircular distribution. More generally, the only time the operator $p (L_\\nu L_\\mu) - q L_\\mu$ has orthogonal polynomial eigenfunctions is when $\\mu$ and $\\nu$ are related by a Jacobi shift."},"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":"0909.1097","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-09-06T17:45:45Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"1c698dc54e93fd1f0c174b82542546af2a648c9c029da46556cc2248b0bf78b3","abstract_canon_sha256":"8d1993200c317300a4a5d47a83de2801ac5acebe0c4620644879ad28829c568f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:13.329974Z","signature_b64":"mFFSnSGZgycZMAI2FpCEgHdaEa6FFCbbJ10q25zgzBSSeFQjrJDmRz5WgEEGmXKyDgVeoHDMMKbog/JzEoqiDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"142d616dad0a980ae3c16ad09b8cf8ad7a1e1b1d96808636ee7216fc1372b7e8","last_reissued_at":"2026-05-18T04:20:13.329420Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:13.329420Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bochner-Pearson-type characterization of the free Meixner class","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.CO","authors_text":"Michael Anshelevich","submitted_at":"2009-09-06T17:45:45Z","abstract_excerpt":"The operator $L_\\mu: f \\mapsto \\int \\frac{f(x) - f(y)}{x - y} d\\mu(y)$ is, for a compactly supported measure $\\mu$ with an $L^3$ density, a closed, densely defined operator on $L^2(\\mu)$. We show that the operator $Q = p L_\\mu^2 - q L_\\mu$ has polynomial eigenfunctions if and only if $\\mu$ is a free Meixner distribution. The only time $Q$ has orthogonal polynomial eigenfunctions is if $\\mu$ is a semicircular distribution. More generally, the only time the operator $p (L_\\nu L_\\mu) - q L_\\mu$ has orthogonal polynomial eigenfunctions is when $\\mu$ and $\\nu$ are related by a Jacobi shift."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1097","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0909.1097","created_at":"2026-05-18T04:20:13.329501+00:00"},{"alias_kind":"arxiv_version","alias_value":"0909.1097v1","created_at":"2026-05-18T04:20:13.329501+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.1097","created_at":"2026-05-18T04:20:13.329501+00:00"},{"alias_kind":"pith_short_12","alias_value":"CQWWC3NNBKMA","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"CQWWC3NNBKMAVY6B","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"CQWWC3NN","created_at":"2026-05-18T12:25:59.703012+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/CQWWC3NNBKMAVY6BNLIJXDHYVV","json":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV.json","graph_json":"https://pith.science/api/pith-number/CQWWC3NNBKMAVY6BNLIJXDHYVV/graph.json","events_json":"https://pith.science/api/pith-number/CQWWC3NNBKMAVY6BNLIJXDHYVV/events.json","paper":"https://pith.science/paper/CQWWC3NN"},"agent_actions":{"view_html":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV","download_json":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV.json","view_paper":"https://pith.science/paper/CQWWC3NN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0909.1097&json=true","fetch_graph":"https://pith.science/api/pith-number/CQWWC3NNBKMAVY6BNLIJXDHYVV/graph.json","fetch_events":"https://pith.science/api/pith-number/CQWWC3NNBKMAVY6BNLIJXDHYVV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV/action/storage_attestation","attest_author":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV/action/author_attestation","sign_citation":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV/action/citation_signature","submit_replication":"https://pith.science/pith/CQWWC3NNBKMAVY6BNLIJXDHYVV/action/replication_record"}},"created_at":"2026-05-18T04:20:13.329501+00:00","updated_at":"2026-05-18T04:20:13.329501+00:00"}