Algebraic independence results for the sixteen families of q-series

Carsten Elsner; Shun Shimomura; Iekata Shiokawa; Yohei Tachiya

doi:10.1007/s11139-010-9235-4

Algebraic independence results for the sixteen families of q-series

Carsten Elsner, Shun Shimomura, Iekata Shiokawa, Yohei Tachiya

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

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
Pages (from-to)	315-344
Number of pages	30
Journal	Ramanujan Journal
Volume	22
Issue number	3
DOIs	https://doi.org/10.1007/s11139-010-9235-4
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

Access to Document

10.1007/s11139-010-9235-4

Cite this

@article{4eecf9365a2840eba7e3fb00a6b87f0c,

title = "Algebraic independence results for the sixteen families of q-series",

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.",

keywords = "Algebraic independence, Jacobian elliptic functions, Nesterenko's theorem, Ramanujan functions, q-series",

author = "Carsten Elsner and Shun Shimomura and Iekata Shiokawa and Yohei Tachiya",

year = "2010",

month = apr,

day = "29",

doi = "10.1007/s11139-010-9235-4",

language = "English",

volume = "22",

pages = "315--344",

journal = "The Ramanujan Journal",

issn = "1382-4090",

publisher = "Springer Netherlands",

number = "3",

}

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 - 2010/4/29

Y1 - 2010/4/29

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 - Ramanujan functions

KW - q-series

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

AN - SCOPUS:77954863200

SN - 1382-4090

VL - 22

SP - 315

EP - 344

JO - The Ramanujan Journal

JF - The Ramanujan Journal

IS - 3

ER -

Algebraic independence results for the sixteen families of q-series

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this