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We establish two recursive formulae for these functions. We show that the specialization to $q=1$ is a rational function involving the Chebyshev polynomials. We give a closed form for $A_n^{\\ell \\rightarrow \\ell+1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\\ell+1$ and $\\ell=1,2$, we relate the generating series to partia"},"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":"1502.05322","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-02-18T17:56:37Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"126ecb01c9772f2365b7aa3569491225a5e3ec70e36a9fa76ed982fe013572a2","abstract_canon_sha256":"48326fdd1609c02fed7b3756b1071749d07c9f2ee7686f88e2c511e5bf5fa382"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:51.980422Z","signature_b64":"zVKL+jniORGuX9p/lOUeREPqALlNSwqpAm7RHSbnESj3PBC5RziAdXX53UAwBwbQEjVey+BORRENyDgWHTWBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e501a4376342c19b25c9532981dd059f56f56e77cc9785d3c80cf181913dc9bf","last_reissued_at":"2026-05-18T02:26:51.980025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:51.980025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Demazure Flags, Chebyshev polynomials, Partial and Mock theta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Lisa Schneider, Rekha Biswal, Sankaran Viswanath, Vyjayanthi Chari","submitted_at":"2015-02-18T17:56:37Z","abstract_excerpt":"We study the level $m$--Demazure flag of a level $\\ell$--Demazure module for $\\frak{sl}_2[t]$. 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