Algebraic independence results for the sixteen families of q-series

作者：Elsner Carsten^*; Shimomura Shun; Shiokawa Iekata; Tachiya Yohei

来源：Ramanujan Journal, 2010, 22(3): 315-344.

DOI：10.1007/s11139-010-9235-4

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

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.

• 出版日期2010-8