### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Ramanujan Journal*,

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

**Algebraic independence results for the sixteen families of q-series.** / Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata; Tachiya, Yohei.

Research output: Contribution to journal › Article

*Ramanujan Journal*, vol. 22, no. 3, pp. 315-344. https://doi.org/10.1007/s11139-010-9235-4

}

TY - JOUR

T1 - Algebraic independence results for the sixteen families of q-series

AU - Elsner, Carsten

AU - Shimomura, Shun

AU - Shiokawa, Iekata

AU - Tachiya, Yohei

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Algebraic independence

KW - Jacobian elliptic functions

KW - Nesterenko's theorem

KW - q-series

KW - Ramanujan functions

UR - http://www.scopus.com/inward/record.url?scp=77954863200&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954863200&partnerID=8YFLogxK

U2 - 10.1007/s11139-010-9235-4

DO - 10.1007/s11139-010-9235-4

M3 - Article

VL - 22

SP - 315

EP - 344

JO - The Ramanujan Journal

JF - The Ramanujan Journal

SN - 1382-4090

IS - 3

ER -