A remark on Nesterenko’s theorem for Ramanujan functions

Abstract

A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic -subanalytic functions (LA-functions) is given. In [2] Lion and Rolin give an explicit description of functions on R n (n ∈ Z, n > 0), called by them LE-functions, defined as finite compositions of globally subanalytic functions with logarithmic and with exponential functions. This enables them to obtain the fundamental results of van den Dries, Macintyre and Marker [1] without making use of model theory. One important step in their study is their Preparation Theorem for LA-functions. To quote this theorem we first recall some basic definitions from [2]. If F is any family of real functions on R n , a subset E of R n is called an F-set if E = i∈I j∈J E ij where I and J are finite and for each (i, j) ∈ I × J, E ij = {φ ij > 0} or E ij = {φ ij = 0} or E ij = {φ ij < 0}, with some φ ij ∈ F.

Algebraic independence results for the sixteen families of q-series

“…], (vii), (viii), (xv), and (xvi) in [5, Lemmas 3, 1, 2, and 4, resp. ], and (vi), (ix), (x), (xi), (xii), and (xiv) in [6,Lemmas 9,4,5,7,8,and 6,resp.]. We have only to prove (v) and (xiii).…”

“…Proof The values in (i) are given in [7,Corollary 2]. Here, we exhibit the method taking (ix) as an example.…”

“…The first three A 1 (q), A 3 (q), and A 5 (q) with q = x 1/2 define the Ramanujan functions P (x), Q(x), and R(x), respectively. Applying Nesterenko's theorem, we derived the following theorem [7,Theorem 2]: For any algebraic number α with 0 < |α| < 1, the numbers A 1 (α), A 2i+1 (α), and A 2j +1 (α) with 1 ≤ i < j and (i, j ) = (1, 3) are algebraically independent over Q. Furthermore, A 7 (q) = A 3 (q) + 120A 3 (q) 2 as q-series.…”