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

##### MSC:
 11J82 Measures of irrationality and of transcendence
##### Citations:
Zbl 0147.30901; Zbl 0984.11037
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