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
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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 -