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Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$.\n  Our proof relies entirely on $q$-series combined with the Bailey pairs The original"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.15107","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-14T17:24:36Z","cross_cats_sorted":[],"title_canon_sha256":"eb210e76ba2c013a9f47bfc19970cce8bc602d505642930de3aba064d1dd96fc","abstract_canon_sha256":"f591061b8f09fc7848a46c5cb420d6a5e02acdbb5cacb891bbdc56841be90612"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.2","canonical_sha256":"5818c6199dd0a04b5c8b301dc79190757c7eb5741f589403a551e3280297de77","last_reissued_at":"2026-05-17T21:57:19.108916Z","signature_status":"unsigned_v0","first_computed_at":"2026-05-17T21:40:25.776754Z"},"graph_snapshot":{"paper":{"title":"Solutions for Hecke Sum Questions of Banerjee and Bringmann","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A two-variable refinement with parameter a of the Hecke sum for two-color partitions is proved using only q-series and Bailey pairs.","cross_cats":[],"primary_cat":"math.NT","authors_text":"George E. Andrews, Mohamed El Bachraoui","submitted_at":"2026-05-14T17:24:36Z","abstract_excerpt":"The present authors introduced a two-color partition series $S(q)$ and conjectured a Hecke-type formula for the even part of $(q^4;q^4)_\\infty S(q)$. Banerjee and Bringmann proved the conjecture by using indefinite theta functions, modular completions, and Sturm's theorem. They also asked whether a direct proof, for instance one based on Bailey-type ideas, could be found, and they suggested that the odd residue classes may be worth studying. We prove a two-variable refinement with an additional parameter $a$.\n  Our proof relies entirely on $q$-series combined with the Bailey pairs The original"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove a two-variable refinement with an additional parameter a. Our proof relies entirely on q-series combined with the Bailey pairs. 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