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Measures of linear independence for values of entire transcendental solutions of certain functional equations. II. (Maße für die lineare Unabhängigkeit von Werten ganz transzendenter Lösungen gewisser Funktionalgleichungen. II.) (German) Zbl 1041.11050

Consider the function \[ f(x)=\sum_{n=0}^\infty\left(\prod_{\nu=1}^nA(\nu)\right)^{-1}x^n \] where \(A(n)=R_1(n)q_1^n+\cdots+R_r(n)q_r^n\). Under certain assumptions, the authors obtain the linear independence measure \[ | h_0+h_1f(a_1)+\cdots+h_lf(a_l)| \geq C_1\exp(-C_2(\log H)^{2(r+1)/(r+2)}) \] for all integers \(h_0,\ldots,h_l\) with \(0<\max| h_i| \leq H\). As examples, the measure applies in the following situations: With \(r=1\), for \(f(x)=\sum_{n=0}^\infty x^n/R(1)\cdots R(n)q^{n(n+1)/2}\) where \(q\) is an integer not equal to \(0,\pm1\), \(R\) is in \({\mathbb Q}[x]\), \(R(n)\neq0\) for all integers \(n\geq0\), \(a_1,\ldots, a_l\) are non-zero rationals, and no \(a_i/a_j\) with \(i\neq j\) is a power of \(q\). With \(r=2\), for \[ f(x)=\sum_{n=0}^\infty x^n/ \prod_{\nu=1}^n R(\nu)U_\nu \] where \(U_n\) is defined by \(U_0=0, U_1=1, U_n=8U_{n-1}-6U_{n-2}\), \(R\) is in \({\mathbb Q}[x]\), \(R(n)\neq0\) for integers \(n\geq0\), \(a_1,\ldots, a_l\) are non-zero rationals, and no \(a_i/a_j\) with \(i\neq j\) lies in the subgroup of \({\mathbb Q}(\sqrt{10})^\times\) generated by \(4\pm\sqrt{10}\). The particular recurrence in this example is explained by noting that the assumptions required to prove the theorem include the assertions that \(q_1, \ldots, q_r\) comprise sets of conjugate algebraic numbers with just one of maximal absolute value and all divisible by a prime ideal in the ring of integers of \({\mathbb Q}(q_1,\ldots,q_r)\).
For Part I, see ibid. 69, 103–122 (1999; Zbl 0961.11022).

MSC:

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

Citations:

Zbl 0961.11022
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References:

[1] Amou, M.; Katsurada, M., Irrationality results for values of generalized Tschakaloff series, J. Number Theory, 77, 155-169 (1999) · Zbl 0929.11020 · doi:10.1006/jnth.1999.2376
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