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Measures of linear independence of values of entire transcendental solutions of certain functional equations. (Maße für die lineare Unabhängigkeit von Werten ganz transzendenter Lösungen gewisser Funktionalgleichungen.) (German) Zbl 0961.11022
The prototype for the functions in this paper is \(T_q(x)= \sum^\infty_{n=0} q^{-n(n-1)/2}x^n\). Diophantine results for the values of such functions go back to L. Tschakaloff [Math. Ann. 80, 62-74 (1920; JFM 47.0167.02)] who proved the linear independence of 1, \(T_q(a_1), \dots, T_q (a_l)\) when \(q\) is a sufficiently large integer, \(a_1,\dots,a_l\) are rational and none of the ratios \(a_i/a_j\) with \(i\neq j\) is a power of \(q\). J.-P. Bézivin [Manuscr. Math. 61, 103-129 (1988; Zbl 0644.10025)] treated the more general function \[ f(x)= \sum^\infty_{n=0} \left(\prod^n_{\nu=1} A(\nu) \right)^{-1}x^n. \] Here, \(A(n)=\beta_1 q_1^n+ \cdots+ \beta_rq^n_r\neq 0\) for all \(n\) and \(\beta_1, \dots,\beta_r\) are non-zero rational numbers, \(q_1,\dots, q_r\) are multiplicatively independent algebraic integers forming complete sets of conjugates and with the maximum absolute value of the conjugates in each set greater than 1, and such that the ring of integers in the field \(K=\mathbb{Q} (q_1, \dots,q_r)\) has a prime ideal which contains all the principal ideals \((q_i)\). Let \(a_1, \dots, a_l\) in \(K\) be non-zero and such that the \(a_i/a_j\) with \(i\neq j\) and \(q_1, \dots, q_r\) are multiplicatively independent and suppose each \(a_i/ A(n)\) is rational. Then there is an effective linear independence measure of the shape \[ \bigl|h_0+h_1f(a_1) +\cdots+ h_lf(a_l) \bigr|\geq C_1 \exp \bigl(-C_2(\log H)^{2r/ (r+1)}\bigr) \] where the \(h_i\) are integers and \(0< \max_{1\leq i\leq l}|h_i|\leq H\). As an example, the theorem applies to the function \[ f(x)= \sum^\infty_{n=0} x^n/\bigl( \delta^{n(n+1)/2} B(1) \cdots B(n) \bigr) \] where \(B(n)= \gamma^n_1+ \gamma^n_2+ \gamma^n_3\), the \(\gamma_i\) are the three roots of the cubic \(x^3-x^2 -x-1\), and \(\delta\) is an integer different from 0 and \(\pm 1\). The required properties in this case follow from the fact that the \(\gamma_i\) are conjugate PV-numbers. The proof of the theorem itself is based on the Hilbert-Perron-Skolem method.

MSC:
11J72 Irrationality; linear independence over a field
11J82 Measures of irrationality and of transcendence
11J91 Transcendence theory of other special functions
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