Algebraic independence results for the sixteen families of q-series

Carsten Elsner, Shun Shimomura, Iekata Shiokawa, Yohei Tachiya

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


The sixteen families of q-series containing the Ramanujan functions were listed by I.J. Zucker (SIAM J. Math. Anal. 10:192-206, 1979), which are generated from the Fourier series expansions of the Jacobian elliptic functions or some of their squares. This paper discusses algebraic independence properties for these q-series. We determine all the sets of q-series such that, at each algebraic point, the values of the q-series in the set are algebraically independent over ℚ. We also present several algebraic relations over ℚ for two or three of these q-series.

Original languageEnglish
Pages (from-to)315-344
Number of pages30
JournalRamanujan Journal
Issue number3
Publication statusPublished - 2010


  • Algebraic independence
  • Jacobian elliptic functions
  • Nesterenko's theorem
  • Ramanujan functions
  • q-series

ASJC Scopus subject areas

  • Algebra and Number Theory


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