{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:723TENOCEQDTVJYDYCTXRTDEZP","short_pith_number":"pith:723TENOC","schema_version":"1.0","canonical_sha256":"feb73235c224073aa703c0a778cc64cbd730cbd0b308d39b55e6986edbb321ee","source":{"kind":"arxiv","id":"1606.08327","version":3},"attestation_state":"computed","paper":{"title":"A kind of orthogonal polynomials and related identities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2016-06-27T15:48:38Z","abstract_excerpt":"In this paper we introduce the polynomials $\\{d_n^{(r)}(x)\\}$ and $\\{D_n^{(r)}(x)\\}$ given by $d_n^{(r)}(x)=\\sum_{k=0}^n\\binom{x+r+k}k\\binom{x-r}{n-k} \\ (n\\ge 0)$, $D_0^{(r)}(x)=1,\\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\\ (n\\ge 1).$ We show that $\\{D_n^{(r)}(x)\\}$ are orthogonal polynomials for $r>-\\frac 12$, and establish many identities for $\\{d_n^{(r)}(x)\\}$ and $\\{D_n^{(r)}(x)\\}$, especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. As an application we extend rece"},"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":"1606.08327","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-27T15:48:38Z","cross_cats_sorted":["math.CA","math.CO"],"title_canon_sha256":"8098a2f53554abc948cb0f959a65beebaa7ef266bae9431eba21744e0f34ceb3","abstract_canon_sha256":"39b82f064a86cb5bcf29cd4fe47890e06a95b568dfa7e8008173d5b5660184f0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:33.012678Z","signature_b64":"jvQNWXbq0kEa+GjDNJEbNo6Lf+i8v75xp7c80t6U7oQvQTjuG5s3Pse4nd6nYyuYZV+NLZykN2Z74mF+uHnSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"feb73235c224073aa703c0a778cc64cbd730cbd0b308d39b55e6986edbb321ee","last_reissued_at":"2026-05-18T00:30:33.011957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:33.011957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A kind of orthogonal polynomials and related identities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2016-06-27T15:48:38Z","abstract_excerpt":"In this paper we introduce the polynomials $\\{d_n^{(r)}(x)\\}$ and $\\{D_n^{(r)}(x)\\}$ given by $d_n^{(r)}(x)=\\sum_{k=0}^n\\binom{x+r+k}k\\binom{x-r}{n-k} \\ (n\\ge 0)$, $D_0^{(r)}(x)=1,\\ D_1^{(r)}(x)=x$ and $D_{n+1}^{(r)}(x)=xD_n^{(r)}(x)-n(n+2r)D_{n-1}^{(r)}(x)\\ (n\\ge 1).$ We show that $\\{D_n^{(r)}(x)\\}$ are orthogonal polynomials for $r>-\\frac 12$, and establish many identities for $\\{d_n^{(r)}(x)\\}$ and $\\{D_n^{(r)}(x)\\}$, especially obtain a formula for $d_n^{(r)}(x)^2$ and the linearization formulas for $d_m^{(r)}(x)d_n^{(r)}(x)$ and $D_m^{(r)}(x)D_n^{(r)}(x)$. 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