### Abstract

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 language | English |
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Pages (from-to) | 315-344 |

Number of pages | 30 |

Journal | Ramanujan Journal |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 Apr 29 |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Elsner, C., Shimomura, S., Shiokawa, I., & Tachiya, Y. (2010). Algebraic independence results for the sixteen families of q-series.

*Ramanujan Journal*,*22*(3), 315-344. https://doi.org/10.1007/s11139-010-9235-4