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Borwein-Bradley-type identities for Ramanujan-type series for 1/π4

  • Noam Shalev
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Abstract

We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for 1/π4 and π4 with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on x3, x4, x6 or x8. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's 5F4 sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.

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Cite

BibTeXRIS
BibTeX
@misc{shalev2026borweinbradleytype,
  title = {{Borwein-Bradley-type identities for Ramanujan-type series for $1/π^4$}},
  author = {Noam Shalev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04770},
  eprint = {2610.04770},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04770},
}
RIS
TY  - UNPB
TI  - Borwein-Bradley-type identities for Ramanujan-type series for 1/π^4
AU  - Noam Shalev
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04770
UR  - https://arxiv.org/abs/2610.04770
AB  - We prove ten conjectures of Z.-W. Sun on Ramanujan-type series for 1/π^4 and π^4 with harmonic numbers. They follow by comparing Taylor coefficients in new Borwein-Bradley-type identities, whose products depend only on x^3, x^4, x^6 or x^8. These identities come from parametric extensions of the series of Cullen and Zhao. We derive the extensions from K. C. Au's Wilf-Zeilberger (WZ) seeds, and their sums are trigonometric. Our proofs use Guillera's periodicity argument and give new proofs of the original series. In the same way, Dougall's _5F_4 sum gives half-integer analogues of the identities of Koecher, Borwein-Bradley and Cohen, whose Taylor coefficients are Apéry-like series for products of zeta values.
ER  -