Baker-type estimates for linear forms in the values of \(q\)-series. (English) Zbl 1064.11054

Väänänen, Keijo; Zudilin, Wadim

Can. Math. Bull. 48, No. 1, 147-160 (2005).

The authors obtain lower estimates for the absolute values of linear forms in values of certain generalized Heine series (\(\phi\) below) at several non-zero points of \(K\), which is either \(\mathbb{Q}\) or an imaginary quadratic number field. These bounds depend on the individual coefficients in \(O_K\), the ring of integers in \(K\), not only on the maximum of their absolute values. Such estimates were first given, in the case of the classical exponential function, by A. Baker [Can. J. Math. 17, 616–626 (1965; Zbl 0147.30901)].
More precisely, the authors’ main result reads as follows. Let \(q\in O_K, |q|>1, s\in \mathbb{N}\), and \(P\in K[X]\) of degree \(\leq s\) satisfying \(P(0)\neq 0, P(q^{-\nu})\neq 0\) for \(\nu=0,1,\dots \), and define the function \(\phi(z)\) by \(\sum_{n\geq 0} z^n/(q^{sn(n-1)/2}\prod_{\nu=0}^{n-1} P(q^{-\nu}))\). Suppose \(\alpha_1,\dots ,\alpha_m\in K^\times\) with \(\alpha_i/\alpha_j \not \in q^\mathbb{Z}\) for \(i\neq j\), and in addition, \(\alpha_i \neq P_sq^n\) for all \(n\in \mathbb{N}\) in case \(\deg P = s, P_s\) denoting the leading coefficient of \(P\). Then, for any \(\varepsilon\in \mathbb{R}_+\), there exists \(C=C(\varepsilon)\in \mathbb{R}_+\) such that \[ |h_0+h_1\phi(\alpha_1)+\cdots+h_m\phi(\alpha_m)|>C(\prod_{i=1}^m \max(1,|h_i|))^{-\mu-\varepsilon} \] holds for all \((h_0,\dots,h_m)\in O_K^{m+1} \setminus \{\underline{0}\}, \mu=\mu(m,s)\) being a positive constant, explicitly given in terms of \(m\) and \(s\) (satisfying \(\mu(m,s)=13ms+O_s(1)\) as \(m\rightarrow \infty)\). In the particular cases of the Tschakaloff function and of the \(q\)-exponential function slightly better lower bounds are obtained.
The proofs use a variant of Siegel’s classical method applied to a system of Poincaré-type functional equations and the connection between its solutions and \(\phi(z)\) applied already by M. Amou, M. Katsurada and K. Väänänen [Acta Arith. 99, 389–407 (2001; Zbl 0984.11037)]. Another essential ingredient are Padé-type approximations of the second kind for these solutions.

Reviewer: Peter Bundschuh (Köln)

MSC:

11J82

Measures of irrationality and of transcendence

Keywords:

Chakalov function; generalized Heine series; lower estimates; \(q\)-exponential function

Citations:

Zbl 0147.30901; Zbl 0984.11037