{"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. The original even identity and the odd identity then follow as corollaries by letting a=1.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Bailey pair technique applies directly to this specific two-color series and its even/odd parts without requiring modular completions or additional verification steps.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Andrews and El Bachraoui prove a two-variable generalization of the Hecke sum identity for S(q) via Bailey pairs, recovering the even and odd cases as corollaries when a=1.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A two-variable refinement with parameter a of the Hecke sum for two-color partitions is proved using only q-series and Bailey pairs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e9af26003876451b78ad2e4187dc07a5a71cf6478e4f749d29d794ee8986163e"},"source":{"id":"2605.15107","kind":"arxiv","version":1},"verdict":{"id":"edf704da-c6dd-4696-b726-5b0bca953836","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:09:16.578618Z","strongest_claim":"We prove a two-variable refinement with an additional parameter a. Our proof relies entirely on q-series combined with the Bailey pairs. The original even identity and the odd identity then follow as corollaries by letting a=1.","one_line_summary":"Andrews and El Bachraoui prove a two-variable generalization of the Hecke sum identity for S(q) via Bailey pairs, recovering the even and odd cases as corollaries when a=1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Bailey pair technique applies directly to this specific two-color series and its even/odd parts without requiring modular completions or additional verification steps.","pith_extraction_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."},"references":{"count":7,"sample":[{"doi":"","year":1976,"title":"G. E. Andrews,The Theory of Partitions, Addison-Wesley, Reading, MA, 1976; reprinted by Cambridge Uni- versity Press, Cambridge, 1998","work_id":"5786e449-ace0-40f6-a77b-ee57e59c8417","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1986,"title":"G. E. Andrews,q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, vol. 66, American Ma","work_id":"e43466ef-b44f-4940-9553-d1900aadb7f0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"G. E. Andrews and M. El Bachraoui,Congruences for two-color partitions with odd smallest part, arXiv:2410.14190","work_id":"71faf992-d8f5-45a7-a99c-2eda0ba9eb06","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Proof of a conjecture of Andrews and Bachraoui on a Hecke sum","work_id":"b8f55b65-5e02-47b7-9b26-34a7161fdd51","ref_index":4,"cited_arxiv_id":"2605.10300","is_internal_anchor":true},{"doi":"","year":2004,"title":"G. Gasper and M. Rahman,Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Appli- cations, vol. 96, Cambridge University Press, Cambridge, 2004","work_id":"2d742963-570f-48d3-9644-db9382563a5f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":7,"snapshot_sha256":"98676fb8f4e00d7ce8fa66e9fd5bc29a0d034aa01af73f0d0e0f0901931e0528","internal_anchors":1},"formal_canon":{"evidence_count":1,"snapshot_sha256":"223b73e92b6ccdcbcacd129a338c6c332d6e979b8e9c5c7b3455fd3639592933"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}