{"paper":{"title":"Bispectrality and the ad conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Adapted ad-conditions provide a route to new examples in exceptional orthogonal polynomials and non-commutative bispectral problems.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"F. Alberto Grunbaum","submitted_at":"2026-03-01T06:17:41Z","abstract_excerpt":"At the beginning of the study of the bispectral problem, see [18], the ad-conditions played a crucial role in finding non-classical instances. The connection with the ad-conditions has reappeared in several different incarnations of the bispectral problem. Here we show that, properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case. Even at this more advanced stage of the field one may hope that finding explicit solutions of these ad-conditi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That finding explicit solutions of these ad-conditions will provide an additional route to new examples, based on their prior role in the bispectral problem.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Adapted ad-conditions from bispectrality offer a route to new examples of exceptional orthogonal polynomials in commutative and non-commutative cases.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Adapted ad-conditions provide a route to new examples in exceptional orthogonal polynomials and non-commutative bispectral problems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f823a95d119b218eb59c6ad826559b475b87f8bb6dbfbec253202e521827d02d"},"source":{"id":"2603.00939","kind":"arxiv","version":2},"verdict":{"id":"f168bcb2-40d8-4155-bd8c-77d5abbee0b8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T18:49:24.868054Z","strongest_claim":"Properly adapted versions of these conditions, see [44], can play an important role in areas including, for instance, the study of exceptional orthogonal polynomials. This is also true in the non-commutative case.","one_line_summary":"Adapted ad-conditions from bispectrality offer a route to new examples of exceptional orthogonal polynomials in commutative and non-commutative cases.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That finding explicit solutions of these ad-conditions will provide an additional route to new examples, based on their prior role in the bispectral problem.","pith_extraction_headline":"Adapted ad-conditions provide a route to new examples in exceptional orthogonal polynomials and non-commutative bispectral problems."},"references":{"count":52,"sample":[{"doi":"","year":1978,"title":"Adler, M., Moser, J.,On a class of polynomials connected with the Korteweg-deVries equationComm. Math. Phys.611, (1978) 1–30","work_id":"9b49f35b-1418-4c57-8257-d0c7369eaaf6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1977,"title":"Airault, H. , McKean, H. P., Moser, J.,Rational and elliptic solutions of the Korteweg-deVries equation and a related many body problemCommun. Pure and Applied Math3097 (1977) 95–178","work_id":"daa4f12d-ba37-493a-87f3-45f058cff406","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Letters A 222(1996) pp 59-66","work_id":"d952bca4-f651-49d3-ae72-c258947a7d1a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"Bakalov, B., Horozov, E., Yakimov, M.,Bispectral algebras of commuting differentail operatorsComm. Math. Physics190,2, 1997 pp 331-373","work_id":"8e275d24-f668-4614-a57c-7fd60ce943c8","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1949,"title":"Bargman, V.On the connection between Phase Shifts and Scattering Potential, Rev, Modern Physics,211, July 1949 pp 488–493","work_id":"00f9a2fe-d1c2-4322-909c-b2bbab15a24d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":52,"snapshot_sha256":"da87a5ccc1ca735fafb14c5d9f6743cac3ac3458e99373dd79cec6b09dc5cc54","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"0f41c6b5703cf2e28e97a0514b87490e7a258673892def0574149a23b58445d9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}