Pith Number

pith:LAMMMGM5

pith:2026:LAMMMGM52CQEWXELGAO4PEMQOV

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Solutions for Hecke Sum Questions of Banerjee and Bringmann

George E. Andrews, Mohamed El Bachraoui

A two-variable refinement with parameter a of the Hecke sum for two-color partitions is proved using only q-series and Bailey pairs.

arxiv:2605.15107 v1 · 2026-05-14 · math.NT

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

7 extracted · 7 resolved · 1 Pith anchors

[1] G. E. Andrews,The Theory of Partitions, Addison-Wesley, Reading, MA, 1976; reprinted by Cambridge Uni- versity Press, Cambridge, 1998 1976

[2] 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 1986

[3] G. E. Andrews and M. El Bachraoui,Congruences for two-color partitions with odd smallest part, arXiv:2410.14190

[4] Proof of a conjecture of Andrews and Bachraoui on a Hecke sum · arXiv:2605.10300

[5] G. Gasper and M. Rahman,Basic Hypergeometric Series, 2nd ed., Encyclopedia of Mathematics and its Appli- cations, vol. 96, Cambridge University Press, Cambridge, 2004 2004

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-17T21:40:25.776754Z
Last reissued	2026-05-17T21:57:19.108916Z
Builder	pith-number-builder-2026-05-17-v1
Signature	unsigned_v0
Schema	pith-number/v1.0

Canonical hash

5818c6199dd0a04b5c8b301dc79190757c7eb5741f589403a551e3280297de77

Aliases

arxiv: 2605.15107 · arxiv_version: 2605.15107v1 · pith_short_12: LAMMMGM52CQE · pith_short_16: LAMMMGM52CQEWXEL · pith_short_8: LAMMMGM5

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LAMMMGM52CQEWXELGAO4PEMQOV \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 5818c6199dd0a04b5c8b301dc79190757c7eb5741f589403a551e3280297de77

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f591061b8f09fc7848a46c5cb420d6a5e02acdbb5cacb891bbdc56841be90612",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-14T17:24:36Z",
    "title_canon_sha256": "eb210e76ba2c013a9f47bfc19970cce8bc602d505642930de3aba064d1dd96fc"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15107",
    "kind": "arxiv",
    "version": 1
  }
}