### Abstract

We study the algebraic independence of values of the Ramanujan q-series. It is proved that, for any distinct positive integers i, j satisfying (i, j)≠(1, 3) and for any with 0<{pipe}q{pipe}<1, the numbers A_{1}(q), A_{2i+1}(q), A_{2j+1}(q) are algebraically independent over. Furthermore, the q-series A_{2i+1}(q) and A_{2j+1}(q) are algebraically dependent over if and only if (i, j)=(1, 3).

Original language | English |
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Pages (from-to) | 211-221 |

Number of pages | 11 |

Journal | Ramanujan Journal |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Jan |

### Fingerprint

### Keywords

- Algebraic independence
- Nesterenko's theorem
- Ramanujan functions

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Ramanujan Journal*,

*21*(2), 211-221. https://doi.org/10.1007/s11139-009-9163-3

**A remark on Nesterenko's theorem for Ramanujan functions.** / Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata.

Research output: Contribution to journal › Article

*Ramanujan Journal*, vol. 21, no. 2, pp. 211-221. https://doi.org/10.1007/s11139-009-9163-3

}

TY - JOUR

T1 - A remark on Nesterenko's theorem for Ramanujan functions

AU - Elsner, Carsten

AU - Shimomura, Shun

AU - Shiokawa, Iekata

PY - 2010/1

Y1 - 2010/1

N2 - We study the algebraic independence of values of the Ramanujan q-series. It is proved that, for any distinct positive integers i, j satisfying (i, j)≠(1, 3) and for any with 0<{pipe}q{pipe}<1, the numbers A1(q), A2i+1(q), A2j+1(q) are algebraically independent over. Furthermore, the q-series A2i+1(q) and A2j+1(q) are algebraically dependent over if and only if (i, j)=(1, 3).

AB - We study the algebraic independence of values of the Ramanujan q-series. It is proved that, for any distinct positive integers i, j satisfying (i, j)≠(1, 3) and for any with 0<{pipe}q{pipe}<1, the numbers A1(q), A2i+1(q), A2j+1(q) are algebraically independent over. Furthermore, the q-series A2i+1(q) and A2j+1(q) are algebraically dependent over if and only if (i, j)=(1, 3).

KW - Algebraic independence

KW - Nesterenko's theorem

KW - Ramanujan functions

UR - http://www.scopus.com/inward/record.url?scp=75549086655&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=75549086655&partnerID=8YFLogxK

U2 - 10.1007/s11139-009-9163-3

DO - 10.1007/s11139-009-9163-3

M3 - Article

AN - SCOPUS:75549086655

VL - 21

SP - 211

EP - 221

JO - The Ramanujan Journal

JF - The Ramanujan Journal

SN - 1382-4090

IS - 2

ER -